Correction:
"Postman Postman/Do Your Duty" should have been:

Policeman Policeman/ Do Your Duty"

I was surprised and pleased that in the 1980s I heard my daughter and her friends recite "Policelady Policelady". Maybe there were women who were police when I was growing up, but I can't recall any.

I suppose 'do your duty' meant to give the person who can 'wiggle, and wobble and do the split' {or "do the twist"}.

Even though the girls were saying to the police officer "do your duty" by giving the girl a kind of citation or arrest her because she's shaking her derriere too much {according to 'adult proper society', they actually showing off how well they could do those moves.