The films' unique look was achieved by pasting cels inside transparent plastic cups, so that a movement cycle could be achieved by simply rotating the cups. Here's some background production quotes from Paul Fierlinger via a WikipediaLiveJournal posting:
"... instead of cycling them as flat sheets on an animation stand, I could roll them up to look like cups and move them around real household objects – just the way kids do when they play with ordinary objects."
"We ended up cutting the character out of the cels and pasting to the inside of Clear Plastic Dixie cups with Scotch spray (it’s transparent) and ran a test, using Larry's (Larry Gold) house as backgrounds (read “we” as X wife, Helena and two little sons Peter and Philip and little daughter Emily). Dave Thurman and Jim Thurman (Sesame Street writer who became the TLSG writer and V.O.) liked the results and I was given a budget to create 3 spots in 3 weeks (2 minutes each I think, but they might have been longer)."
"I had a couple of months to create the hundreds of cups that were needed to play out any possible scenario and create the sound tracks."
"The composer was Larry Gold. His friend, Stuart Horn; an eccentric, dreamy, sometimes charmingly sleazy and enormously creative gay guy wrote the song’s lyrics. The four-part harmony was sung by a period Rock Star, Singer, Songwriter, and Actor, Ezra Mohawk and John Avarese and recorded on a four-track Teac reel-to-reel machine in Larry’s kitchen on 2nd Street (we are all Philadelphians)."