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For a Mobiüs strip, the bug would find the world had been mirrored along the width of the strip if it made a round trip and came back to the same point. Other bugs that stayed behind would claim that the traveller itself had become mirrored.

(If you do the experiment with paper, the bug would be on the other side of the paper from where it started, but I don't think this concept of "side" exists when the strip is described as a manifold. The bug lives in the strip, it's not walking on top of it. It helps to imagine the strip to be transparent.)

For a Klein bottle the same would happen, but there the bug could also find the world mirrored along a different direction, depending on how it travelled.

I'm pretty sure the property of sidedness comes up when you embed the manifold in euclidian space. The property seems to be related to orientability.

Möbius. Not Mobiüs.

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