Roman Legionaries vs. Cavalry
What do we know about the tactics Roman infantry used against enemy cavalry?
The Romans themselves had an equestrian tradition, though the cavalry arm of their military forces have been (unfairly) branded on account of poor performance at several major battles. That said, even up to the 5th/6th Centuries AD, the backbone of the Roman war machine consisted of infantrymen equipped with broad shields and thrusting swords, trained for close-combat.
The traditional view of history would suggest that the legions were fairly weak when opposing cavalry. Two of Rome's ugliest defeats, Carrhae (53 BCE) and Adrianople (378 CE) were suffered at the hands of enemy horsemen. Roman legions were shot to pieces by Parthian horse-archers on a number of occasions; a legion was destroyed in the East during the reign of Antoninus Pius, at least three entire Roman armies were slaughtered by Persian horsemen in the 3rd Century; and in 36 BCE Marcus Antonius' army was bloodied by Parthian cavalry. Lance-and-sword armed Sarmatian and Germanic horsemen also inflicted a number of defeats on Roman forces; another legion was annihilated by Sarmatian cataphracts in Pannonia in the late 1st Century.
Accounts of decisive Roman victories in the field against Parthians and Persians - though they do exist - are comparatively rare. Roman armies under Trajan, Verus, Severus, Caracalla, Galerius, and Julian the Apostate all penetrated Persia and sacked the capital of Ctesiphon, proving that the Romans were fully capable of crippling their eastern rivals. The legions must have perfected some means of containing the leathal combination of light horse-archers and heavily-armored mounted lancers.
The testudo "tortoise formation", known as the fulcum or foulkion in the Late Empire, was widely used when facing light cavalry. This formation nominally rendered the Roman ranks untouchable, at least as long as they held their position. That said, Roman testudos were shot to pieces at Carrhae as well as at an unnamed battle against the Persians in 233 CE.
The agmen quadratum, also known as the orbis (literally "world") formation was also recorded when fighting cavalry strong opponents. This formation apparently entailed all the centuries or cohorts of a Roman infantry force forming a hollow circle or square, facing the enemy on every side. This obviously took away the great flaw of the testudo (being attacked from behind), while also enabling the legionaries to return enemy missiles with their own javelins.
The traditional panoply for a Roman legionary included a rectangular or oval body shield, a medium-length thrusting sword, a dagger, and one or two heavy javelins. There is evidence to suggest modifications to this gear when facing cavalry-strong opponents, especially in regards to polearms. Arrian, in his Battle Order against the Alans, suggests that half of his legionaries (those in the front ranks) were equipped with thrusting spears (hastae), perhaps something of a miniature sarissa. Only their comrades in the back were equipped with javelins, and they used these to harass the enemy horsemen while their brethren in the front defended the formation with their long spears.
In 215 CE Antoninus "Caracalla" equipped a force of 16,000 Praetorians and legionaries in imitation of Alexander's phalanx. Added to their equipment was a latter-day sarissa, presumably identical to those used by the Makedonians. He boasted to the Parthian monarch Artabanos IV that he had an infantry force "invincible when fighting with spears", and this encouraged the latter to make peace with him. Epigraphic evidence suggests that elements of Antoninus' "phalanx" were still in operation in the East a decade after his murder.
The "tortoise" and "world/hollow square" formations appear to have been favorites when fighting cavalry-strong enemies. Thrusting spears may have replaced the traditional pila for some legionaries in the Eastern army.
What else do we know about tactics utilized by the Romans when fighting cavalry?