The Swiss and Piedmontese coats of arms and flags might both possibly, repeat possibly, have originated as symbols of loyalty to the Holy Roman Empire.
Flags and coats of arms of Switzerland and Piedmont are certainly visually related.
The question of whether they are historically related is complex and has no certain answer.
Part One of Eight: The Holy Roman Empire.
In 962, Otto I the Great, King of the East Franks or of Germany and kKing of Italy or of Lombardy, was crowned Emperor in Rome. From 962 to 1806, everyone who became emperor was also automatically King of Germany and Italy. In 1032 Emperor Conrad II became king of Arles or Burgundy, and from 1032 to 1806 everyone who became emperor automatically became king of Burgundy.
Piedmont was mostly or entirely within the borders of the medieval kingdom of Italy within the Holy Roman Empire, and modern Switzerland is entirely within the lands of the former kingdoms of Germany, Lombardy, and Burgundy within the Holy Roman Empire.
From about 1190 or some, a person elected emperor used the title of Rex Romanorum et semper Augustus "King of the Romans and always Emperor" and after being crowned emperor would use the title Imperator Romanorum et semper Augustus "Emperor of the Romans and always Emperor".
Part Two of Eight: The Imperial coat of arms with the eagle:
There were no national flags nor coats of arms in ancient Roman society, but the eagle of Jupiter was a symbol associated with the Roman Empire. And so the Holy Roman Emperors used eagle symbols, and when heraldry developed in the 12th century (1101-1200) used an eagle as their coat of arms.
At first there was considerable variation in the details of the imperial coat of arms, such as the colors of the background and the eagle, whether the eagle's heads had halos, and how many heads the eagle had.
Writers began describing the imperial coat of arms in works about heraldry in the second half of the 13th century (1251 to 1300). And their usual description of the coat of arms of the King of the Romans was "Or (yellow) a one headed eagle displayed (with wings spread out) sable (black)", while there usual description of the coat of arms of the Emperor of the Romans was "Or (yellow) a two headed eagle displayed (with wings spread out) sable (black)".
By about 1450 (and possibly much earlier) the coat of arms of the King of the Romans was established as: "Or (yellow) a one headed eagle displayed (with wings spread out) sable (black), a halo around the head", and the coat of arms of the Emperor of the Romans was established as: "Or (yellow) a two headed eagle displayed (with wings spread out) sable (black), halos around the heads".
Many principalities, fiefs and city states in the Holy Roman Empire used eagles in their coats of arms, some by coincidence and some to show loyalty to the Emperor.
The dukes and kings of Poland used a white eagle on a red field. The Dukes and Kings of Bohemia used a black eagle on a white field before switching to a white lion on red.
Coats of Arms of the Holy Roman Empire
In the late middle ages a two headed eagle was also used as a symbol in the eastern Roman or "Byzantine" empire, and also by neighboring states. For about 26 years from 1346 to about 1373 Serbian rulers claimed the title of "Emperor of the Serbians and the Romans" and for about 200 years Bulgarian rulers claimed the title of "Emperor of the Bulgarians and the Romans". Those "emperors" and lesser rulers like despots and sebastocrators often used two headed eagle symbols.
And the history of the use of two headed eagles in the Holy Roman empire and the "Byzantine" empire and neighboring realms is quite complex and little known. Accounts of the history of the two headed eagle symbol and coat of arms often have more guesses than proven facts.
Naturally the imperial coat of arms of the Holy Roman emperor was used on the emperor's personal heraldic banner. And the imperial coat of arms and banner were also used to represent the Holy Roman Empire as a whole and loyalty to the Empire.
Part Three of eight: The coat of arms of Swabia with three lions.
The emperors and kings of the Romans from 1136 to 1208 to and from 1212 to 1254 were members of the Hohenstaufen dynasty. In that period the Duke of Swabia was also a member of the Hohenstauffen dynasty, the Emperor or a close relative.
The coat of arms of the Duchy of Swabia is also yellow with black charges, in this case three lions passant (walking) in pale (stacked vertically) sable.
The princely family of Waldburg uses a coat of arms of "or, three lions passant sable langued and armed (with tongues and claws) gules (red) upon a chief (a bar at the top) gules an orb or". And this coat of arms was obviously designed to show their loyalty to the Dukes of Swabia.
But much more powerful rulers also used similar coats of arms with three lions passant in pale.
The coat of arms of Denmark is blazoned as:
Or, three lions passant in pale azure crowned and armed Or langued gules, nine hearts Gules.
The oldest known depiction of the insignia dates from a seal used by King Canute VI c. 1194. The oldest documentation for the colours dates from c. 1270.3
Historically, the lions faced the viewer and the number of hearts was not regulated and could be much higher. The "heart" shapes originally represented waterlily pads; a royal decree of 1972 still specifies these figures as søblade ("lake leaves").
The current design was adopted in 1819 during the reign of King Frederick VI who fixed the number of hearts to nine and decreed that the heraldic beasts were lions, consequently facing forward
The Danish coat of arms was adopted when the king of Denmark was sometimes a vassal of the emperor. Thus it has been suggested that the coat of arms was inspired by the coat of arms of Swabia, or even adopted to show loyalty to the emperor.
King Richard I of England was captured by Duke Leopold of Austria returning from the Third Crusade, and imprisoned by the Duke and Emperor Henry VI until he paid a king's ransom for his freedom. It has also been claimed that Richard had to do homage to the Emperor and hold all of his lands, including England, as a fief of the empire.
After release, Richard adopted a second great seal for the kingdom of England, including a new coat of arms with three lions passant in pale.
Richard I (1189–1199) used a single lion rampant, or perhaps two lions affrontés, on his first seal,6 but later used three lions passant in his 1198 Great Seal of England, and thus established the lasting design of the Royal Arms of England.5 In 1177, his brother John had used a seal depicting a shield with two lions passant guardant, but when he succeeded his brother on the English throne he would adopt arms with three lions passant or on a field gules, and these were then used, unchanged, as the royal arms ('King's Arms') by him and his successors until 1340.6
And it has been speculated that Richard's new coat of arms was inspired by the coat of arms of Swabia, or even designed as a sign of loyalty to the emperor.
Part Four of Eight: The imperial flag with a white cross on a red field.
But the Holy Roman Empire had another symbol in the middle ages.
The Reichsfahne (Imperial flag) was a field ensign of the Holy Roman Empire, originally an equestrian flag or gonfalon. An early bearer was Werner I, count of Winterthur, who carried the flag for Conrad II and Henry III and who died in the battle at Brůdek in 1040. In the 12th century, the Reichsfahne apparently[clarification needed] showed a white cross on a red field. It was the sign of the united armed forces of the Empire until the late 15th century, but it could be sent by the king to local lords to sanction them in their defense of Landfrieden. Thus, king Sigismund gave the banner to the Swiss Confederacy, sanctioning their war against the Habsburgs in 1415.
In the late medieval period, the cross design of the Reichsfahne was replaced by the Imperial eagle. It was treated as an Imperial fief traditionally granted to Swabian nobles. In 1336, it was granted to Ulrich III, Count of Württemberg. On this occasion, it was first referred to as the Reichssturmfahne ("Imperial War Flag"). It remained part of the heraldic insignia of the House of Württemberg until the 19th century. The flag itself was kept in Stuttgart until 1944, when it was destroyed in a bombing raid. The flag showed the imperial eagle in a square field, with a red Schwenkel (pennon) on top. It is not to be confused with the Reichsrennfahne, granted to the Electors of Saxony in their function as Reichserzmarschall. This latter flag showed two crossed swords in a black and white field.2
I don't know how accurate that Wikipedia article is, but certainly a white cross on red was a symbol associated with the Holy Roman Empire.
Since the Holy Roman Empire claimed to be the rightful government of the whole world, both the coat of arms with the two headed eagle and the white cross on red can be considered to be symbols of the whole world and every place in it.
Part Five of Eight: Other flags with a white cross on a red field.
During the 12th and 13th centuries, armies of the Ghibelline communes usually adopted the war banner of the Holy Roman Empire —white cross on a red field—as their own. Guelph armies usually reversed the colors—red cross on white. These two schemes are prevalent in the civic heraldry of northern Italian towns and remain a revealing indicator of their past factional leanings. Traditionally Ghibelline towns like Pavia, Novara, Como, Treviso and Asti, continue to sport the Ghibelline cross. The Guelph cross can be found on the civic arms of traditionally Guelph towns like Milan, Vercelli, Alessandria, Padua, Reggio and Bologna.
The flag of Denmark is called the Dannebrog and is often considered the oldest national flag in the world.
A banner with a white-on-red cross is attested as having been used by the kings of Denmark since the 14th century.2 An origin legend with considerable impact on Danish national historiography connects the introduction of the flag to the Battle of Lindanise of 1219.3
The Danish national anthem claims the first Dannebrog fell from heaven during the battle of Lindanise. There are many theories about the origin of the Dannebrog. Since the Dannebrog looks a lot like the Reichsfahne of the Holy Roman Empire, it is possible there is a connection, and so it has been speculated that the design of the Dannebrog might have been adopted to show loyalty to the Holy Roman Empire since many of the early Danish kings were vassals of the emperor.
The coat of arms of Switzerland is defined by a 2017 law as:
The 2017 law defines the "Swiss coat of arms" (German: Schweizerwappen) formally the "Coat of Arms of the Swiss Confederation" (German: Wappen der Schweizerischen Eidgenossenschaft, French: armoiries de la Confédération suisse consistent, Italian: stemma della Confederazione Svizzera) as "a Swiss cross in a triangular shield". The Swiss cross (Das Schweizerkreuz, la croix suisse, la croce svizzera) is defined as
"a white, upright, free-standing cross depicted against a red background, whose arms, which are all of equal size, are one-sixth longer than they are wide."9
The Swiss cross originates as a field sign worn by the troops of the Old Swiss Confederacy. Use of the emblem is attested with certainty in the context of the Old Zürich War, for the year 1444, when the Tagsatzung defended itself against allegations that the troops of the Confederacy had deceptively used two different field signs (Heerzeichen). Aegidius Tschudi (Chronik II.390) cites a song containing this allegation, specifying that the Swiss were bearing "two kinds of crosses, white at the back and red in front" (Si trügend zweierlei Crützeren, [...] Hinden wiß und vornen rot) to deceive the Zürich side.2
Heraldic representation of the Confederacy was by representations of the cantonal coats of arms. Petermann Etterlin on the title page of his Kronika of 1507 shows the coats of arms of the Thirteen Cantons (besides those of Chur, Valais and St. Gallen) surrounding the imperial coat of arms, reflecting the claim of the cantons of the Confederacy to imperial immediacy within the Holy Roman Empire.
The first known example where the cantonal coats of arms are shown as surrounding a Swiss cross representing the Confederacy is a medal commissioned by the Tagsatzung from Zürich goldsmith Hans Stampfer in 1547 as a gift for the French princess Claude. The obverse side of this medal shows the coats of arms of the Thirteen Cantons in their order of precedence, the reverse shows the coats of arms of the Associates, both groups surrounding a central cross with ornamental foliage. Similar representations are found throughout the early modern period, on commemorative medals and on regimental seals used by Swiss Guards in French service.3 The first mention of the Confederate Cross (Eidgenossen Crütz) shown in a shield (on pennies minted by Schwyz) dates to 1533.4
The white cross has been used as the field sign (attached to the clothing of combatants and to the cantonal war flags in the form of strips of linen) of the Old Swiss Confederacy since its formation in the late 13th or early 14th century.
The ultimate origin of the white cross is attributed by three competing legends: To the Theban Legion, to the Reichssturmfahne (Imperial War Banner) attested from the 12th century, and to the Arma Christi that were especially venerated in the three forest cantons, and which they were allegedly allowed to display on the formerly uniformly red battle flag from 1289 by king Rudolph I of Habsburg at the occasion of a campaign to Besançon.
Thus there is a theory, which might possibly be correct, that the Swiss cross coat of arms and flag is based on the Reichsfahne or Reichssturmfahne of the Holy Roman Empire and might have been adopted to show loyalty to the Empire.
Part Six of Eight: The coat of arms of Savoy.
According to legend, the original coat of arms of the Counts of Savoy in the Kingdom of Burgundy was "or, an eagle displayed sable", thus closely resembling the coat of arms of the Holy Roman Empire. The later counts and dukes of Savoy used a coat of arms of "gules (red) a cross argent (white)". Sometimes those coats of arms were displayed together, with the eagle having a shield on its torso with the coat of arms with the cross.
Obviously the old coat of arms of Savoy that closely resembles the imperial coat of arms could have been assumed to show loyalty to the Empire, and the new coat of arms of Savoy, with the same design as the Reichsfahne might have also been assumed to show loyalty to the Emperor.
Part Seven of Eight: Piedmont.
Piedmont is a region in northwestern Italy, at the foot of the Alps, that was ruled for centuries by the House of Savoy.
the flag of Piedmont is one of the official symbols of the region of Piedmont, Italy. The current flag was adopted on 24 November 1995.9
The flag of Piedmont is essentially the arms of the Prince of Piedmont, the title for the eldest son of the King of Sardinia.2 When Duke Amadeus VIII of Savoy gave his eldest surviving son the title of "Prince of Piedmont" in 1424, he added a heraldic label to the coat of arms distinguish it from the general coat of arms of the House of Savoy.3
Two versions exist of the Piedmontese flag: one with a blue border, and one without a blue border. The latest version, officially adopted in 1995, features a gold fringe and orange ribbon,4 although the flag is sometimes seen without it.5
So the coat of arms of the Prince of Piedmont and the region of Piedmont was simply the coat of arms of the Duchy of Savoy with a blue label for difference.
Part Eight of Eight: Conclusion.
And it is quite possible that the coats of arms and flags with white crosses on red fields of Denmark, Ghibeline cities in Italy, Switzerland, and Savoy might have been adopted to show loyalty to the Holy Roman Empire with its Reichsfahne with a white cross on red.
But I don't think it will ever be possible to know for certain.
Added 03-22-2020. the coat of arms of the City of Vienna also has a white cross on red, whether it has any connection with other coats of arms or flags with the same design.