The Möbius strip, topological equivalence, and one-way functions

A donut and a coffee mug are topologically equivalent because one can be transformed into the other without cutting, tearing, or adding more holes. It's easier to understand this if you imagine both shapes being made of soft clay or Play Dough. 

However, a Möbius strip and a loop strip are not topologically equivalent even though they have the same number of holes. The first has one side and the other has two sides. A Möbius strip made of clay could be molded into a loop strip, but not the reverse is not possible. 

It seems there are topological transformations that are not reversible. Thus, they are analogous to one-way functions.