I'm trying to wrap my head around the concept of orientability as an intrinsic property of a manifold. Assume I'm in some (3-dim) manifold for which I'd like to decide its orientability; what could I do in that case? I guess this setting is sufficient to avoid thinking with immersions/embeddings.
My attempts so far:
Comparing to the 2D case. If a 2D object travels along the Möbius strip, it would appear flipped (left and right) when it's back to its original location.
But that still sounds abstract. What would I perceive if I travel around in a 3-manifold and come back? Maybe things suddenly look mirrored, I guess, but not sure. And I see no reason why it has to be left-right (instead of upside-down, since that's also perpendicular to my travel curve).
I did find some hints in another question; the above is similar to the idea of the second answer there. Another idea is to observe counter-clockwise-ness, mentioned in the comments of the first answer. Neither of them is sufficiently detailed to make me understand, so I have to ask this separate question.
Thanks in advance.