> If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post).
Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).
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> If you try to label one side as up, and then you move continuously around the Möbius strip, you'll end up labelling that same side as down!"
This statement is true (though it is using a different definition of "up" and "down" than you are), and your claim that spinning the strip will show a contradiction with it is false. See my post downthread in response to your "spin the strip as you would a wheel". I phrased it there in terms of which edge is the "outside" and which is the "inside" edge, but the same is true of "up" vs. "down" sides relative to the strip (note that I emphasized that phrase in my other post).
Note that the issue is not that spinning the strip does not define an "inside" (towards the axis) vs. "outside" (away from the axis) in the 3-space in which the strip is embedded; of course it does (and similarly for "up" and "down", if you want to use those terms instead, in the embedding 3-space). The issue is that there is no way to use this to construct a continuous orientation on the strip itself, the way you could on an orientable surface, such as an ordinary ring (a disk with a hole cut in the center).