The reason I ask is because of the widespread use of the gladius during this time by the infantry, which as we know the short length had many advantages for. But surely a short sword isn't great for mounted combat, so I would think the spatha would be ideal.
But I can only find claims that the spatha found widespread use later on as foreign cavalry units that already used longswords continued to do as after becoming auxilii and introduced it, so to speak. My second guess would be spears but it seems the Romans weren't particularly interested in them except as projectiles such as the pila.
Thank you in advance to anyone taking the time to indulge me :)
Woo, question times! So to start with, I'll give you a quick rundown of the Roman cavalry and their general use et alla. From there, I'll discuss what we know (and what we don't) about their equipment.
The Romans were pretty well known for two things, among many others: having a nigh-unstoppable infantry in the melee, and having really poor cavalry. Even authours who are trying to talk the Roman war machine up are forced to admit, a la Polybius, that Roman cavalry were exemplary for bravery, if nothing else. Wew know that the early Roman cavalry, in the early and middle Republic at least, were equipped in the "Greek fashion." Beyond this utterly unhelpful note by Polybius, we really know nothing much about their equipment. Scholars have theorized that this included a sword for the close melee and a handful of javelins for skirmishing. Recognizing this weakness, the Romans, with their infinite love of Stealing Other Peoples' Stuff, drew the vast majority of their cavalry from auxiliary regiments, from people who were known for their horsemanship especially. We start seeing significant examples of this during the Second Punic War, both for and against the Romans.
Hannibal has a ball kicking Roman tail up and down the peninsula, and this was partly aided by the clear superiority of his Numidian cavalry, widely recognized as the best horsemen of that period. Going back to that earlier delightful tendency of the Romans, Scipio (later Africanus) wooed the Numidians away from the Carthaginians, using them to good effect at Hannibal's final defeat at the Battle of Zama. A quick summary of the battle - the infantry got into an all out slugfest, only absolutely resolved when the Romans (Well, Numidians really), having crushed the Carthaginian cavalry, flanked and routed Hannibal's elite core of veterans.
A more valuable question, therefore, would be about the cavalry fighting under the Roman banner, rather than the Roman cavalry themselves. The Numidians are described as pulling feats on horseback that mirror the Mongols - flipping underneath their horses to hamstring enemy cavalry, "Parthian-style" hit and run tactics with light javelins, and so forth. So your answer as to their equipment would be more of a light cavalry core focused around agility and skirmishing until the foe was wounded, tired, and discouraged, then closing for a close-range rout. The Gauls, over a century later under (and against) Caesar are described similarly - light cavalry, focused on agility over armour. Heck, Crassus' disastrous defeat at Carrhae is partly blamed on his lack of any cavalry to counter the enemy cavalry - and the only kind of cavalry that could take on horse archery to any degree would be the same aforementioned light skirmishers.
So that's a quick down and dirty of your Republican era stuff, but what happened when Augustus (may he live forev- oh wait) completely restructured the Roman army in the late First Century BC? Well, he made the army pretty institutionalized. Infantry were mostly heavy, with auxiliary cohorts of skirmishers added on as needed, but often just using auxiliary heavy infantry to complement the Romans. Cavalry, on the other hand, was almost exclusively devoted to the auxilia. So your question about the spatha is a little self explanatory: the Roman cavalry adopted the spatha thanks to the auxiliaries...because the Roman cavalry was basically always auxiliary. We've actually excavated some first c. AD spatha from Scotland. The spatha was usually a longer sword, about 26-36 in (65-91cm), imitating the gladius in style. Other weapons, though, included a few small javelins for throwing and one "fighting spear." Gravestone depictions often show cavalrymen with a spear in hand. Regulation equipment, according to a text recording of weapon inspections in first century AD Britain, was one fighting spear, one sword, and two subarmales (throwing javelins and such). Josephus (1st c. AD Palestine) confirms this, talking about Roman cavalry with one fighting spear and multiple javelins stored in a quiver.
Other options for more specialized cavalry could be more focused on shock (12 foot! lances), or heavy power (yeah, the Romans stole the cataphracts), or even horse archers, but these mostly start in the second century AD.
Hope this answers your questions!
Edit because further reading: Check out Goldsworthy's *The Complete Roman Army - it's a very accessible and readable work that succinctly covers a huge amount of information, much of it on this (and similar) topic.
Follow up question:
what happened to a cavalryman who lost his horse during the march. Was he incorporated into the infantry or the auxiliary or was it "below his status"? If not, did the horseless guy continue to hang out with the legion or was he perhaps given a leave to find a new mount?