I can’t answer this for all of Rome, during all of Roman history, from the founding to the fall of Constantinople, but I can tell you that during the late Republic and early Principate, Roman cavalry were explicitly trained to mount from both sides of the horse. Horsemanship was not central to Roman culture as a whole, as it was to some of their eastern enemies. They were not common farm animals or pets. Their role was predominantly military. Roman ideas about cavalry were mostly inherited from the Greeks, or imported from enemies and subjugated peoples. In this era, after the Marian reforms professionalized the military, Roman cavalry was overwhelmingly non-citizen Auxiliaries from territories with stronger horse cultures, like Spain. Doctrinally, they drew heavily from Xenophon, who insisted cavalrymen be able to mount quickly from either side.

To meet the case in which the horseman may chance to be leading his horse with the left hand and carrying his spear in the right, it would be good, we think, for every one to practise vaulting on to his seat from the right side also. In fact, he has nothing else to learn except to do with his right limbs what he has previously done with the left, and vice versa. And the reason we approve of this method of mounting is (8) that it enables the soldier at one and the same instant to get astride of his horse and to find himself prepared at all points, supposing he should have to enter the lists of battle on a sudden.

Note that the expectation is that at least one hand is full when mounting. This is important, because they weren’t mounting they way modern riders typically do, by stepping up with a stirrup. Stirrups did not exist. Roman cavalrymen were trained to jump onto their mount’s back, fully armed and armored. If that sounds impossible, well, it wasn’t easy, but you have to remember that Roman cavalry horses of this period were usually not very large. The types were not standardized, but were drawn from the region where the units were raised. This usually meant Gaul, where small, sturdy horses similar to modern Icelandics dominated, or Spain, where lighter mounts similar to modern Barbs were archetypical. And the Romans trained this skill. The modern gymnastics event of vaulting is derived from the Greco-Roman training method which used a wooden horse. In the fifth century CE, Publius Flavius Vegetius Renatus wrote in “De Re Militari”,

“To Be Taught to Vault: The ancients strictly obliged both the veteran soldiers and recruits to a constant practice of vaulting. It has indeed reached our times, although little regard is paid to it at present. They had wooden horses for that purpose placed in winter under cover and in summer in the field. The young soldiers were taught to vault on them at first without arms, afterwards completely armed. And such was their attention to this exercise that they were accustomed to mount and dismount on either side indifferently, with their drawn swords or lances in their hands. By assiduous practice in the leisure of peace, their cavalry was brought to such perfection of discipline that they mounted their horses in an instant even amidst the confusion of sudden and unexpected alarms.

Flavius here laments a perceived deterioration in this aspect of training, but acknowledges it still exists, even in the late Western Empire. Likewise, when describing the qualities of an ideal Decurion, the cavalry equivalent to a Centurion, he says,

In like manner the Decurion is to be preferred to the command of a troop for his activity and address in mounting his horse completely armed; for his skill in riding and in the use of the lance and bow; for his attention in forming his men to all the evolutions of the cavalry; and for his care in obliging them to keep their cuirasses, lances and helmets always bright and in good order.

So mounting skills are important enough to get equal billing with other horsemanship and military skills.

This ambidextrous approach to training was not exclusive to mounting. Cavalry tactics of this period were dominated by missile weapons, and a charging troop would normally wheel to the right, releasing their javelins, so their shields, which they always carried in their left hand, would be facing the enemy as they withdrew. Shock tactics were not unknown to the Romans, including the use of large two handed lances, but heavy cavalry was really only just starting to play a significant shock role at the very end of this period, with the introduction Cataphractii under Hadrian. However, since you could not count on the tactical situation favoring this right wheel, troops trained to wheel in both directions. A major part of Roman cavalry training was the figure eight, with flying changes. Horse and rider were trained to change directions, at speed, switching which front foot led, by repetition, riding in figure eights until it became rote. Lead changes are something horses do naturally, for comfort and agility, but they usually have to be taught to do it with a rider, on command. This ability to turn with equal agility, speed and comfort was a big tactical advantage. Julius Caesar, who was unusually skilled in handling cavalry for a Roman general, used a left wheel with his Gallic cavalry at Ilerda. So we see a pattern developing here, stressing tactical flexibility. Something as peripheral as which side you wore your sword on, and its effect on the ease with which you mounted, was not a major concern.

As for the sword itself, it was not a major cavalry weapon. In fact, in the Republican era, cavalrymen carried the same short gladius as legionaries. That would not likely be a huge impediment to mounting anyway. For most Roman cavalry, their main weapons were javelins, and greek style spears with pointed metal butts, which could (tactical flexibility again) be wielded in either direction without swapping grip, and continue to be useful even if the tip were broken off. They did also field mounted archers and slingers, albeit not in the numbers of their eastern opponents with cavalry centric armies, like the Parthians.

So no, for purposes of mounted warfare, the Romans did not mount from the right. I am very skeptical that the convention of mounting from the left actually developed from sword wearing preferences at all, even in the age of stirrups. The difficulty of swinging a sword leg over a horse is just not that great compared to the tactical utility of ambidextrous mounting. I don’t even fight from horseback, but I taught my horse to stand still for mounting from both sides, simply for safety’s sake. I can’t count on always having the opportunity to do things the conventional way. The Romans agreed, at least from the cavalry perspective (and during the late Republic and early Principate).

War Horse: A History of the Military Horse and rider, Louis A. DiMarco

Roman Cavalry Training and the Riding School of the Mauretanian Horse Guard, Michael P. Speidel

On Horsemanship, Xenophon

De Re Militari, Publius Flavius Vegetius Renatus

EDIT: Found a better translation of Xenophon.