Great post by Aurelian, including the key phrase for any infantry of any period when facing heavy cavalry: "provided they were arrayed properly".

Catch a Roman force in marching columns or otherwise unprepared for combat and they will be in deep trouble, perhaps not as much as some infantry since their discipline and good corp of underofficers might enable them to rally and form a defensive line, but they will take more casualties then they would if ready for battle.

Even a homogeneous force of Legionnaires using just their typical equipment and prepared for battle will be difficult for the cavalry to beat. Pila wouldn't be as good as longer, heavier spears for receiving a charge, but they could be used in this role. Even better, the rear ranks could throw their pila a few seconds before the charge hit home, attempting to disrupt the cavalry and thus reduce their impact. Depth of ranks - and the discipline to hold them - would be used to absorb the charge, slowing and halting the cavalry within the Roman lines, unless they can break through or away. The Roman system of shuffling their squads around to reinforce weakened areas of the line will give them a good chance of making sure the cavalry don't push through.

So how to beat them if you're the cavalry? If you have horse archers they might be able to disrupt the Roman lines enough that your heavy horse can successfully charge, providing the Romans in turn don't have their own missile troops. Assuming just heavy cavalry vs Legionnaies, then perhaps a series of small feinted charges with part of the cavalry force, attempting to roll back the Roman defences by getting them to expend their Pila, leaving them more exposed for the main cavalry charge. The problem would be if the Romans hold their nerve & refuse to waste their Pila, or balancing losses among the decoys against the mass of numbers needed to successfully perform the final charge. A better tactic might be to shadow the Romans, waiting for an opportunity to attack, using the horses' greater mobility to range ahead & prepare obstacles, traps and scout for potential ambush sites.