This notional gadget is about as useful as Congress. But it is interesting.

I couldn’t find much information about Möbius gear trains on-line, so I included an excerpt from the WikiPedia entry for Möbius strip.

gearsThe Möbius strip or Möbius band (pronounced may-be-us) is a surface with only one side and only one boundary component. It has the mathematical property of being non-orientable. It is also a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.

A model can easily be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip together to form a single strip. In Euclidean space there are in fact two types of Möbius strips depending on the direction of the half-twist: clockwise and counterclockwise. The Möbius strip is therefore chiral, which is to say that it is “handed”.