Wikipedia has some good examples, but I am not sure the Möbius strip is nothing more than a novelty or an oddity. Please, why study the Möbius strip?
Scott Mavers Möbius strips did have a practical application back in the days of tape recording: tape with a half-twist (continuous loop tapes) could double the playing or recording time without having to stop and run the tape in the other direction (even when stop and rerun was automatic, there would still be a small temporal discontinuity). Also typewriter ribbons with a half-twist could use ink from both sides of the ribbon and not have to be replaced as often. Some contemporary higher-tech applications are mentioned here:
https://badw.de/fileadmin/members/R/3685/Academy__Presentation_Oct_08_2015.pdf
Scott Mavers Möbius strips did have a practical application back in the days of tape recording: tape with a half-twist (continuous loop tapes) could double the playing or recording time without having to stop and run the tape in the other direction (even when stop and rerun was automatic, there would still be a small temporal discontinuity). Also typewriter ribbons with a half-twist could use ink from both sides of the ribbon and not have to be replaced as often. Some contemporary higher-tech applications are mentioned here:
https://badw.de/fileadmin/members/R/3685/Academy__Presentation_Oct_08_2015.pdf
Scott Mavers
I dont know the answer, but check this out for your why question. It would not answer your question but would explain much more.
https://www.youtube.com/watch?v=36GT2zI8lVA
The Mobius strip is a fundamental structure in the study of surfaces. You'll find many surfaces can be constructed from it. It's the simplest non-orientable surface.
https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Related_objects
https://en.wikipedia.org/wiki/M%C3%B6bius_strip#Applications
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