These are variations on the Truckin' Basic. They are largely equivalent calls for the lead and follow, respectively. There are some small stylistic differences based on gender norms of the 1930s.
Birdie in a Cage
Language Note: "Bird", in this context, was slang for a woman
Follow marks time in place
Leader trucks around follow counter-clockwise one complete circle
Birdie in a Cage Rhymes
Words that rhyme with Cage: age, stage, page, wage, engage, sage, rage, guage, …
Rhymes for Birdie in a Cage:
They say that all the world is a stage, well put on a show, call it birdie in a cage.
Prowling around is an old gray cat, birdie in their cage ain't having that
Organ Grinder Swing
Leader squats/takes a knee
Follow trucks around leader counter-clockwise one complete circle
Organ Grinder Swing Rhymes
Words that rhyme with Swing: bring, cling, ding, fling, king, ring, sing, sting, thing, wing, …
Rhymes for Organ Grinder Swing:
Leads take a knee, pull out a ring, put your hands together for the Organ Grinder Swing