heavy cavalry engaging heavy cavalry

[Timianus]

Cavalry has always played a very secondary role in the classic-era warfare, but gained greater importance in the late antiquity, especially when the Eastern Roman Empire had to fight against Sassanians and Central Asian steppe nomads.
In many battles, engagements between light cavalry with heavy infantry, heavy calvary with heavy infantry, light cavalry against heavy calvary were described, yet there seemed to be relatively few engagements between heavy cavalry against heavy cavalry.

When such a situation happened, what was the typical tactic?
DId the horsemen charge at each other and penetrate each others' ranks with the lances and swords, like in the movies, with the opposite fractions often charging pass each other to regroup of the other side?
Or did the horsemen charge at each other in close formation, and then slow down to a trot when they closed in, where the front ranks would engage in hand-to-hand combat, like in heavy infantry?
Or would one side always try to charge the other on the flanks and the rear, doing everything to avoid head-to-head confrontation?

IIRC, Constantine defeated Maxentius' heavy cav. by parting his lines, allowing them to ride through the center, and then pulling his own wing cavalry to the back to surround them. I remember reading that he equipped his horse with iron-tipped clubs to beat the opposing cataphracti. I can't remember what battle this was though.