The equipment of the legionary horsemen, equites legionis, was, it seems, very similar to that of the auxiliary cavalry troopers based on what pictorial evidence there is. There is little evidence in terms of archaeology, although a number of pieces of cavalry sports armor have been found in legionary contexts and the adlocutio by Hadrian mentions legionary horsemen carrying out what might have been the "Petrine Throw" as described as part of the hippika gymnasia in Arrian's Taktika -- which would imply similar training and equipment.
The few tombstones we have of equites legionis do not give us much in terms of the type of helmet worn (the best preserved is from Bonn and is, unfortunately, bare-headed) but various auxiliary types are most likely and shields are generally oval or clipped oval types. The sword is usually the spatha, but early on, say in the 20s A.D., the gladius might still have been carried (as show on the Arch at Orange).
One thing that would be that their commander was likely one of the high-ranking supernumerary centurions attached to the legion -- you could do a head swap to a helmet with the transverse crest to make them stand out -- rather than a tribune as previously thought. Other officers know include the optio and vexillarius, who seem to have been part of the turma-like sub-units of legionary cavalry (there is no known legionary decurio so the sub-units were not turmae per se). By the way, they were not "attached to the legion," but were drawn from the men already serving. They remained. for administrative purposes, on the rolls of their century although they had their own barracks.
I would also rate them highly in wargames terms since they were not scouts and messengers as so often stated, but rather served as the legate's bodyguard unit on the battlefield and provided members of his personal bodyguard, the singulares legati legionis.