My Scottish-English mom from Oklahoma sang the “horsey” version of this to me between 1947-1955. She sang it to the tune of, “Did you ever see a Lassie?” At school, kids would stand in a circle, holding hands. We would swing our hands backs and forth, keeping in time with the music. On,”...this way and what-away,” we would turn to the kid on our right, then on our left. Boys would bow and girls would curtesy. This is a charming version https://m.youtube.com/watch?v=Hn4JL7UE42A. The Horsey lyrics fit perfectly.