Can the Möbius strip help "visualize" physics GR or particle physics for a layman?

Factual Questions
1

I was fiddling with one the other day, imagining the piece of paper as so “flat”/"not there, and just generally goggling at it yet again and trying to think out what I could think out about it.

Then I was thinking about the Sysiphean attempts by the physicists on this board to help conceptualize space-time curvature at or near singularities.

Eg, How do the numbers (the 2-D plane) vary from positive to negative consistently and satisfactorily on its “frame of reference” to the satisfaction of mine? They don’t. But they must, I’d wager, to mathematicians. No doubt undergraduates. But me–hmmm…

Anyway, I just thought–in the same way you kind of feel “there must be a moral there” when you hear a story–that perhaps some minute inkling of thought could be inkled out with the proper coaching.

2

I don’t know about GR, but I can believe it can describe spin in QM

On the moebius strip you must go around twice to return to your starting place, a spin-1/2 particle must “rotate” twice before returning to it’s original state.

3

The Möbius strip is sometimes used as an analogy for some aspects of quantum mechanical spin, but I don’t think it’s a very useful one. By the time you finish saying what it is you’re trying to describe, you’ve already described it, and you don’t need the analogy any more.

It is not particularly relevant to anything in GR.