I understand that if you were to walk on the surface of a Möbius strip you would have the same perspective as if you walked on the outer surface of a cylinder. However, would it be possible for someone to determine whether they were on a Möbius strip or cylinder.
If we imagine that we’re walking on a broad walkway, and that we can’t peek over the edge, either at the lateral surface or to the other side, then I don’t think there’s a way to tell. Suppose the walkway has a handrail, on “both” sides, and you start marking the handrail as you keep your right hand on it. After completing a full loop, you’ll be underneath the point across the walkway from where you started. From there, you won’t see any mark yet made, because you’re underneath the path on which you started. If you keep going, you’ll eventually, after another loop, return to the point where you started, and the only mark you see will be on your handrail, not on the handrail across the road to your left.
What has happened here is clearer if you consider what happens when we cut a mobius strip along its midline. Making that cut adds a second edge, producing a normal loop, one edge of which is the original edge of the mobius strip, and the other edge of which is produced by the cut. The walk described in the above paragraph is a walk along one edge of the resulting loop.
If we can reach down and mark the lateral edge of our path, then there’s a way to tell. We make regular marks (or a continuous mark) along the lateral edge to our right, and occasionally we check across the path on our left, to see if any marks are on the lateral edge of the path there. Halfway along the complete (2-loop) walk, you would notice marks on the left side, made from “underneath” the path. That would be evidence that you’re on a mobius path.