>>12691859Conway's Zip Proof (
https://www.maths.ed.ac.uk/~v1ranick/papers/francisweeks.pdf) or Stillwell's Classical Topology and Combinatorial Group Theory have what you're looking for, although in the context of classifiying topological 2-manifolds.
The gist is that you can decomopose any surface as a sphere with handles or crosscaps attached; a mobius strip is homeomorphic to a sphere with a crosscap. The chief complication is that you can't imbed an unorientable surface into R^3, the way you can a torus. So, cross caps tend to look a bit weird.
