Since you're counting, Ylle, I thought I'd look for some less common types of wheeled things.
This was my first idea, "The Caisson Song", aka "The Caissons Go Rolling Along" or "The Field Artillery Song", written in 1908 by US artillery First Lieutenant Edmund Gruber, a descendant of Franz Gruber, composer of "Silent Night" (which I'm guessing is the opposite of what happens when a fleet of caissons are rolling along - what with all the loud and strong shouting and hey-hey-hee-ing mentioned in this song).
From Wikipedia:
A caisson (US: /ˈkeɪsɒn/) is a two-wheeled cart designed to carry artillery ammunition;[3]^ the British term is "ammunition wagon".
In 1917, John Philip Sousa, mistakenly thinking it was a Civil War era song, modified and marchified it into the "US Field Artillery March".
In 1956, Harold Arberg updated the lyrics, dropping the now obsolete caissons, and it was adopted as the US Army's official song, and renamed "The Army Goes Rolling Along", aka "The Army Song".
This would have been a good twofer, since it's a march and includes the phrase "counter march".
