The Romans only wore something that looked like a toga or kilt whereas the Huns and Sarmatians (an Iranian group who were their allies) wore PANTS to facilitate riding a horse. When we see the art of the Huns/Sarmatians, we see horses quite a bit, and we see that they had amazing prowess on horseback. One of these Iranian groups around 2000 years ago invented the "Parthian Shot" which is shooting an arrow behind while the horse is going forwards.
Anyways, I don't get the impression that horse back riding, charioteering, and cavalry were that big of a deal to the Romans. As a matter of fact, I know for a fact that in 378 AD, the Romans didn't make much use of cavalry during their horrifying defeat at Adrianople.
So is horseback riding and charioteering and cavalry a more associated with South West Asian culture?
The Romans certainly had horses and made very heavy and effective use of their cavalry all throughout their history. Adrianople followed the course it did precisely because the Roman cavalry was poorly lead and was able to be defeated in detail. An individual battle isn't a good comment on a formation as a whole precisely because these mistakes in leadership can occur.
On a broader level, if you look at Trajan's Column you will see that Roman cavalry did, in fact, wear pants.
Horse ownership was absolutely a big part of Mediterranean culture. Before the Romans, the Greeks rode and trained horses for both war and pleasure. Not only that, modern horsemanship theory comes largely from Greek thinking (specifically Xenophon) and not Southwest Asian thinking. Meanwhile, the term "Equestrian" referred to a class of Roman aristocracy and military leaders.
The difference, though, is largely that of geography. Horses require a lot of space, and Greece and Rome are not known for having large grasslands. Therefore, to keep a horse, you had to have either land or a lot of disposable income, which made horse-ownership less practical for the average Roman. But, that does not mean that the horse was not held in high esteem within the culture.
