We used to play “chinese jump rope” too in the 60s at our elementary school in northern British Columbia, Canada. I can’t for the life of me remember what we called this game but it certainly wasn’t “chinese jump rope”. The rhyme we used with it was a version of “Flying Purple People Eater”, not the whole song. It started like this:

One-eyed one-horned
Flying purple people eater
Bees wear
Don’t stare
Flying purple people eater

The game was played in the manner that Eirene from Australia described. If anyone knows the rest of the verse I’d love to hear it.