Is this accurate?
Yes.
From a Euro-centric perspective of history the terms "left and right" originates from the French National Assembly, where unlike the Estate General people weren't divided into estates (some sort of caste system including clergy, nobility and peasantry), but due to the fact that they were initially all 3rd class, grouped according to shared opinions. So closer opinions and associations lead to closer proximity in the sitting order.
Which revealed broader factions such as monarchists sitting more on the right side of the room and the more revolutionary sitting more on the left side of the room. For comparison, previous the clergy had been seated to the right of the podium, the nobility to the left and the 3rd class faced it (the king and ministers being at the podium:
La première séance, le 5 mai, est présidée par Louis XVI en personne, le clergé s'assied à la droite du trône, la noblesse à gauche, le tiers état en face. Les orateurs sont le roi, le garde des Sceaux, Barentin, et le ministre des Finances, Jacques Necker. Wiki
They iterated through quite some of those assemblies often keeping this factioning and sitting order, though not necessarily the people or groups sitting there. So roughly conservatives right, moderates center, reformers left, but who is what could change depending on who sits there and in times of rapid change what is reformist now can be conservative later.
So it apparently took until the 3rd republic left and right to gain an ideological meaning with the left being social democrats waving red flags and the right being reactionaries waving white flags.
So essentially you have 2 left right spectra, one is about categorizing the current political factions with respect to how closely aligned or competing they are with each other and on top of that you have attempts trying to find more universal political spectrum of ideas. Where you have contrastive pairs such as:
liberal vs authoritarian
socialism vs capitalism
egalitarian vs elitist
open minded vs dogmatic
pragmatic vs ideological
cautious vs optimistic
progressive vs conservative
secular vs religious
pluralism vs uniformism
individuality vs conformism
individualism vs collectivism
...
Each of them can be used to span it's own spectra in which you could pick a left or a right wing. Sociology's best guess for a universal left-right-spectra is currently one concerned with the question of social hierarchies. Where "the left" would strive towards an absence of social hierarchies and thus an equal individual freedom, coexisting pluralism and social solidarity. Roughly matching the claims of the French revolutionaries "Liberté, Egalité, Fraternité" (liberty, equality, brotherhood). While "the right" is in favor of social hierarchies and would strive or at least defend or tolerate discrimination, inequality, social darwinism and peer-group concerned politics.
Though WHY "the right" would do so could vary drastically, it could be propagated as "the natural order" (some sort of racism), "the god given rule" (theocracy or rulers legitimized by a religious narrative), it could be the result of a "fair" competition (capitalism), it could be military supremacy (nationalism, imperialism) and so on.
So just because the structure of society looks hierarchical, doesn't mean that the motivation for that is consistent among these groups. So they might agree on a strong leadership by a selected few, but they might vigorously bicker about the selection process.
While on the left if you want equality, liberty and solidarity you kinda have to find the balance between not subjugating yourself to others and not subjugating other to yourself. You can't disallow individual self-interest without enslaving people, but unrestricted individual self-interest also enslaves people.
So on the broader left right spectrum concerned with social hierarchies democracy is a pretty lefty idea. Politics as the ideal of collective self-governance where every voice matters equally (at least in theory) is pretty damn left. We seldom think of it like that because it became the new normal, when monarchies fell out of favor, where political decision making was the job of a king, their advisers and the feudal aristocrats who implemented it or who made political decisions of their own.
Though at least from the idea that is much more hierarchical than considering politics a "republic" (res publica = matter of the public). Now you could argue how current politics implements that ideal and whether it's still largely elected temporal aristocrats or whether it's almost direct participation if people would like to, but the grand narrative had changed, from firmly right wing, to firmly left wing.
Fast forward to the 20th century and the entire idea of such grand narratives would collapse somewhat. Like sure you had those counter-revolutionary and reactionist movements already. Like France had emperor Napoleon succeeding the republic, Europe as a whole had the conservative roll back with the Congress of Vienna, but after WWI and the wave of democratization seemed to have finally won and then Fascism apparently kinda felt anachronistic.
People to this day struggle to make sense of it ideologically and often times end up defining it more by it's actions than by it's ideas.
Like I started saying "yes." (they are far right). Because they undoubtedly are. None of the fascist regimes made it a secret that they strived for absolute power, the Nazis in particular made it clear that they did not consider people as equal, they were overtly racist in particular antisemitic, they were overtly social darwinist (T4 program, https://en.wikipedia.org/wiki/Euthanasia ), they subjugated every aspect of social, political and economical life to their party and said party was overtly hierarchical, like from their earliest youth organizations to the party itself you had military rank systems. It might have been in the beginning less hereditary than the monarchies (unless you belonged to the "wrong" "race" (which is also hereditary or at least not within your means)), but social hierarchies were undeniable. And they weren't just the result of something and ignored, but they were actively introduced and propagated. The dictatorship, the disdain for democracy, the cult of personality and leadership where fundamental aspects of that ideology.
So where on the political spectrum of democratic countries the "right wing" parties might be anywhere from moderately left to center right, so still very left due to their support for democracy, but maybe more focused on their own peer group or national self-interest over cooperation and the interests of other.
The Nazis were right wing, not just relative to the left, which at the time sill included communists, anarchists, social democrats seeing themselves as revisionist marxists and so on, so that wouldn't have been hard, but right wing on the absolute left-right spectra. They were overtly in favor of social hierarchies, of dictatorships, strong leaders, racism, discrimination, inequality, social darwinism, anti-democratic, imperialist, uniformistic rather than pluralistic and so on. Now again at the time they were also less of an outlier in that regard with several other right wing parties also being anti-democratic and monarchistic. Weimar Germany is often quoted to be a democracy without democrats (or at least without enough of those) given that the right and left wanted a different system, though despite doing so for very different reasons they nonetheless strained the system from day 1 and cultivated a pretty heated political climate with street fights as a mode of political interactions not being such an outlier.
The obvious "but" is the "socialistic" part in NSDAP name. The second would be the prosecution of Christian clergy. The third - public works program and other examples of state involvement in the economy.
The "socialism" of the Nazis is very very very different from how economics uses socialism. Theoretically the idea of socialism is kind of applying the ideas of democracy and republics to the economic sphere, so the workers would own the means of production like the citizens own their country and would be able to participate in that democratically. Now what that means in practice can vary drastically depending on how you implement a democracy, whether it's direct or representative and so on. Different topic entirely. Though the conceptual idea is to empower the workers and disperse the authoritative power of the owners over their employees and make that more egalitarian.
The Nazis did none and intended to do none of that. Their ideology wasn't socialism but nationalsocialism seeing that as one word is important because what they aimed for instead was to invoke the feeling that "We're all part of the team, (except for those Jews and Slackers)". So where socialists advocated for class warfare and the workers seeing themselves as an oppressed class fighting united for equality (economic, social and political) or at least better working and living conditions, the Nazis literally advocated for "race warfare". So they had no problem praising the workers for their heroic service for the "common good" of their race warfare, but other than symbolic praise they had no intention to decrease or even level the socio-economic or political gap between the classes on the contrary, again hierarchies everywhere was there thing.
And that's for the German/Arian read workers, for those outside that category it was slavery including murder by labor, like how the prisoners of Auschwitz apparently interpreted the "Arbeit macht frei" at the gates:
Arbeit macht frei (Work makes you free)
durch Krematorium Nummer drei. (Through crematorium number three)
So the "nationalsocialism" is not socialist in the economic sense and not even a general workers movement. In fact the NSDAP was more or less a middle class party which often had very little working class representation
(German links)
https://www.bpb.de/shop/zeitschriften/apuz/archiv/531298/wer-verhalf-der-nsdap-zum-sieg-neuere-forschungsergebnisse-zum-parteipolitischen-und-sozialen-hintergrund-der-nsdap-waehler-1924-1933/
https://www.deutschlandfunk.de/untersuchung-der-mitgliederstruktur-ueberraschende-100.html
They also weren't keen on being a mass movement but Hitler constantly tried to keep his party elite and stopped entry when it became popular.
Their "national socialism" was rather a this "völkischer Nationalismus", so "people's nationalism" where he tried to create the narrative of one fellowship with one goal and that goal is an "us or them" race to the bottom with "The Jews".
Though despite the name not fitting the agenda and action, it might have nonetheless been deliberate given that they actually tried to appeal to workers, with rather limited success as well as to nationalist, so they basically tried to present themselves as representation of all germans and thus used all of the popular buzzwords, though often doing some insane mental contortions to make them fit.
Their actual "left wing" which might have taken that serious was murdered pretty early on during their reign (1934), so their importance stretches more to the period of getting power not towards the Nazi ideology or actions which they were no longer a part of.
With respect to the religion. They did have cooperation with the pope, who apparently hoped it would save his priests if they don't resists, though it didn't. And the Nazis worked on neo-pagan esoterics as some sort of replacement religion to cooptate religion as well. https://en.wikipedia.org/wiki/Thule_Society
Also the thing is "religion" is not a good arbiter in the left right categorization. Just take Christianity before and after it became a state religion of the Roman empire. Often enough a liberation gospel can turn into an appeal to authority once the people practicing a religion went from oppressed minority to oppressive minority/majority. Not to mention that the pure concept of absolute obedience to a god and unquestioned authority makes "religion" interesting for dictators even if they are otherwise atheist.
And the "involvement in the economy" is some sort of an American outlier of people actually pretending to believe in Anarcho-(or rather unregulated)-Capitalism, elsewhere the question isn't so much if you get involved in the economy but how and to what end. For the Nazi progaganda the economy wasn't an ideological concern rather than a pragmatic one in their racist and imperialist goals. Also the Nazi regime is the first to have mass privatized public assets and again they supplied companies with cheap slave labor. Though pretty early on they moved towards a war economy and those are more controlled by the warring government who creates the largest demand.
Though if you look at U.S. propaganda at the time they were also advocating resource efficiency and conservation, rather than a free market and care free consumerism. So it's not that you could take the war time economics as emblematic of a countries economic ideals, those are more often driven by necessities.