In recent (100 years?) usage in the US Army, the name used for this routine is "Sound Off."

About the only thing "standardized" is effectively a "chorus:"

Drill Master: SOUND OFF
Troop: ONE TWO
DM: SOUND OFF
Troop: THREE FOUR
DM: CADENCE COUNT
Troop: ONE TWO THREE FOUR,
Troop: ONE TWO (one-step pause) THREE-FOUR

(Some may omit the "skip-step" in the last line, and do the count in regular left-right-left-right sync.)

Although there are many common "verses" a majority of them are (claimed as) unique to a particular squad, company, or post (or sometimes a local barroom). Quite probably more than half of the better known ones are blatantly OBSCENE, RUDE, AND OFFENSIVE (intentionally).

This CADENCE (it's proper name, it's not a song here) is most generally used on military installations only during the first week of a "boot camp" since it seldom takes the CO's wife more than a week to complain about the obscene shouting and the CO orders the troops to "don't do that anymore" (at least stateside). Occasionally the order will be "don't do that while in the housing area," and one was known to have said "where my wife can hear it."

ARMY lore claims that the AIR FARCE uses "Little Willies" as their usual CADENCE verses, often retaining the "Sound Off" call and response as chorus, since they're "just too nice." This rumor is unconfirmed by my experience. (They may not need marching cadences much, since they usually get a bus to take them to and from where they keep their toys?)

Scouting and other "pickups" on the CADENCE are likely to be badly corrupted, but are mostly harmless and sometimes good for a giggle.

John