What is the 'rank switching' done by Roman Legionnaires in HBO's 'Rome' called, and did it actually happen?

I believe the rotation in battle is commonly accepted - most sources I've read reference it, although my research in Roman history is mostly secondary and tertiary sources, so I'm not an authority. With that as a preface/caveat:

Wikipedia to the rescue

wikipedia 1 describes using the intervals between troops to execute a refresh and support continuous fighting.

"When the first line as a whole had done its best and become weakened and exhausted by losses, it gave way to the relief of fresh men from the second line who, passing through it gradually, pressed forward one by one, or in single file, and worked their way into the fight in the same way. Meanwhile the tired men of the original first line, when sufficiently rested, reformed and re-entered the fight. This continued until all men of the first and second lines had been engaged. This does not presuppose an actual withdrawal of the first line, but rather a merging, a blending or a coalescing of both lines. Thus the enemy was given no rest and was continually opposed by fresh troops until, exhausted and demoralized, he yielded to repeated attacks."[30] same source

Another article references the same concept (I suspect the primary sources are the same, but sometimes different secondary sources can clarify interpretations)

Finally, the Romans had a practical system for the passage of lines, and preferred to reinforce or replace tired units with fresh ones rather than maximizing the depth of the initial fighting line. 1

Just to be clear the reference given in that quote is to " Philip Sabin, "The Roman Face of Battle," Journal of Roman Studies 90 (2000), p. 5

Quoting from Sabin,

However, the primary purpose of the multipleline system seems to have been to allow fresh troops to replace or reinforce tired ones in the front line itself. Livy describes this process clearly for the manipular legion (8.8), albeit in an anachronistically early context, and Caesar speaks of fresh cohorts replacing tired ones at Ilerda and Pharsalus (BC I.45-6, 3.94).28 The ancient authors repeatedly state that it was this advantage of having fresh men fighting tired ones which gave the Romans such an edge over opponents who were in equal or greater overall numbers, but massed in a single fighting line (cf. Livy 9.32, 34.14-I5; Onasander 22). Hence, any model we might develop of Roman infantry combat must be one in which having fresh troops in the fighting line matters at least as much as the physical and psychological advantages of greater formation depth. The second and related point is that our model must be one in which the physical passage of lines to accomplish this line relief would have been a feasible proposition. Scholars have long debated the practicalities of the famous 'chequerboard' deployment of Roman legions, addressing such intractable issues as how wide the gaps between maniples or cohorts would have been, whether and how these gaps were closed before combat, and how they were opened again to allow an engaged first line to admit or withdraw through its supports without making itself catastrophically vulnerable to enemy penetrations in the process.29 It is very hard to find satisfactory answers to these questions, and so any model of infantry combat at the level of individual soldiers which makes it easier to understand how the line relief process may have worked will be much more convincing as a consequence. "

The Romans would have a tactic of three lines, where first the the second and then the third line would press themselves between the first line when needed to let the first line get a breather and reform.

When the first line as a whole had done its best and become weakened and exhausted by losses, it gave way to the relief of fresh men from the second line who, passing through it gradually, pressed forward one by one, or in single file, and worked their way into the fight in the same way. Meanwhile the tired men of the original first line, when sufficiently rested, reformed and re-entered the fight. This continued until all men of the first and second lines had been engaged. This does not presuppose an actual withdrawal of the first line, but rather a merging, a blending or a coalescing of both lines. Thus the enemy was given no rest and was continually opposed by fresh troops until, exhausted and demoralized, he yielded to repeated attacks.

The formation was called "Acies triplex", which just means "triple line". I can't find any reference to a name of the "rank switching".

Source: Lt. Col. S.G. Brady, The Military Affairs of Ancient Rome and Roman Art of War in Caesar's Time