I thought I'd turn my comment into a proper answer. My main source is Adrian Goldsworthy's The Complete Roman Army, a must-read/have for anybody who's interested in that topic, IMO.
Originally, as @SJuan76 linked to, cavalry in the "Polybian" (i.e. pre-Marian reforms around 100 BC) Roman army was the charge of those who could even afford the cost of a horse and its and the riders equipment. These were the equites, who formed about 300 out of a legion of 4,200.
The 300 were divided into 10 turmae and each of these was led by a decurion ("leader of 10"). Unfortunately, Polybius only stated that the cavalry fought with "Hellenistic style equipment" and assumes the reader knows what that means. The estimate is that these we close order units
armed with spear and sword and protected by helmet, cuirass and
However, at this time, cavalry riders had neither saddle, nor spurs, nor stirrups. Riding in that way is a challenge in any case, but fighting was a whole other thing.
Yet, other empires managed, and the Romans found out the hard way the cost of over-focusing on their infantry and neglecting their mounted units. One was Cannae against Hannibal, the other Carrhae against the Parthians, both fielding larger and far superior cavalry units. The result was some of the worst defeats the Roman armies ever suffered.
Still, the Romans continued to use their cavalry mostly as skirmishers and scouts. With the Marian reforms this reached its "conclusion" when the professional army virtually did away with the cavalry (and light infantry) leaving the Roman legionnaire of common imagination. The role of the cavalry (still skirmishing and scouting) was handed to those Alae who had proven to be good at it: e.g. Numidians and Celts. IIRC, I.Caesar also was fond of his Germanic mounted units.
I think @Tom Au has a very good point in stating that, coming from a hilly, even mountainous area, cavalry doesn't become as dominant compared to the people from steppes or desert. Yet, we have mention of Cretan cavalry units as core auxiliaries - a rocky, hilly island!
However, it's important to note that even after the Marian reforms, a Roman soldier would have had to own and provide for their own horse! Even if a common legionnaire in the new professional army would not have had to provide for their own armour. This puts a severe limit on the number of cavalry: only the wealthy, e.g. the equites could afford that expense.
Goldsworthy makes a point that very little is known of the allies of this period, so that we cannot tell to what extend these were trained and disciplined. This in the context of how the auxiliares, such as
the Gallic, German and Spanish cavalry, and the Numidian, Cretan and German skirmishers
became the main supplement to make up for the purely infantry make-up of the legions. Meaning, we have no idea how those supporting allies would have afforded their horses and providing for them.
At least some of those units remained essentially the personal followings of a tribal war-leader, fighting for him in the same way they would have done in inter-tribal warfare. Under Augustus and his immediate successors, the auxilia were turned into a much more regular and professional force.
The cavalry was organised into something called quingeniary (500 strong) and milliary (1,000 strong) forces. Contrary to infantry they were called alae instead of cohortes. A cavalry quingenary ala had 16 turmae of 32 men for a total of 512. Milliary units had 768(!) in 32 turmae.
That's actually nothing to sneeze at in numbers compared to a legion. But the role remained support, scouting and skirmishing.
There is a whole chapter on Equestrian Officers under the Principate and its not easy to pick what to quote.
The equestrians became a more important order than under the Republic, specifically to obtain their loyalty to the emperor instead of any senatorial family. Augustus even created equestrian governors. It's likely valid to ask just what connected these equestrian to actual cavalry, but the fact remained that they were "wealthy enough people to afford and ride horses and their provender".
Goldsworthy states that
there were never more than a dozen or so of these prestigious units in existence, so that such commands were reserved for the ablest, or best-connected, officers.
That does not mean that any of them actually sat, never mind fought, on horseback.
By this time the four-horned saddle also was around and Goldsworthy maintains that, contrary to some historians, this made up for the lack of stirrups when mounting charges or doing much of anything beyond harassment or skirmishing.
However he doesn't explain why the Romans, who were extremely adaptable in all other military matters, then never raised the profile of their cavalry, even when their enemies used it to increasingly devastating effect.
On a purely personal note, speaking as an equestrian myself, I think he is wrong here: the difference between having stirrups or not, regardless of the type of saddle, makes a huge difference in the effectiveness of a mounted soldier. Not just in the ability of the massive charge of the armoured knight from centuries later, but in the ability to be supported when shifting or leaning sideways and a four-horn saddle can restrict turning the upper body (such as the parting shot of mounted steppe archers).
Of course, in late antiquity the Roman army was largely made up of foreign troops, many of them fully mounted, anyway.
In summary, the Romans apparently never made the step, during professionalisation of their military, to also centralise the raising of horses and training them and the mounted soldiers. They left it to those who could afford it, whether their nobility or their allies.
Italy isn't ideal for raising horses, but other parts of the Empire certainly were - e.g. Hispania, Cappadocia (which became the main source of horses during the Eastern Roman/Byzantine Empire).