Is it possible to determine if you were on a Möbius strip? 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.
 A: Clearly the two surfaces are not homeomorphic, but from the perspective of the individual on the strip itself, I'm fairly certain it would be identical, though I'm not sure how to prove the general case. Both have 0 curvature so that's a no go, and no matter what type of markings you make, there is no way to differentiate having a single edge compared to two on a cylinder without allowing some type of puncture or extra dimension normal to the plane's surface. If you mark the ground using colors A and B at the edges, the spots where A overlaps B only happens on "opposite" sides, or where the surface is extended through the twist in connecting the sides, meaning you will have the perception of following two edges like on a cylinder.
A: Leave a piece of bacon on the strip, and then walk around the strip, and then eat it. If your body processes it like normal, then you are on a cylinder. If your body treats like a reversed-chirality strip of bacon, you are on a mobius band.
A: If you drive long enough all of a sudden you notice that everybody is driving on the wrong side of the street!
A: Let's talk about "flat" cylinders and Mobius strips, which means if you drew a triangle, the angles would add up to 180 degrees (versus curved space like a sphere or a Pringles(R) chip, where the sum of the angles are either more than or less than 180 degrees, respectively).  Paper loops will give flat space, but it is worth thinking about it in a slightly different way:
A way to imagine this kind of space is like in the game Asteroids, where if you go off the screen you end up on the other side.  A cylinder is a game of Asteroids where you can't go off the top or bottom sides, but if you go off the right side, you appear on the left side at the same height.  A Mobius strip is the same, but when you go off the right side at height $y$, you appear on the left side at height $1-y$ (where the screen is $1$ unit high).
Light travels in straight lines.  As a flatlander if you look in the $x$ direction, you would see the back of your head whether or not the space is a cylinder or a Mobius strip.  However, the back of your head will be reversed if it is a Mobius strip.

A: The question is not well defined to me. What do you mean by determine, what is available?
You could mark your starting point, start walking, and after you have gone 360 degrees you either see your marking (cylinder) or you don't (Mobius strip).
So, your question is equivalent to whether you can determine how many degrees you have rotated. If you have an external reference point, this is easy to do.
Another way, you could use a long band, white on a side, and white on the other side. You walk, leaving the band behind you, white face up. When you come back to the starting point, if it is still white, you are on a cylinder, otherwise, you are on a Mobius strip.
A: If you can't see "both sides" of the surface simultaneously from any point on the strip then I question whether it really is topologically a Möbius strip.
When I make a "Möbius strip" from a strip of paper, for example, writing on one side does not show through. But the physical strip of paper that I held in my hands and whose ends I taped together has two faces separated by a non-zero distance; in reality, its topology is much more like a torus than like a Möbius strip.
If we add the constraint that we cannot travel or communicate in any way over the edge of the paper from one face to the other, each short segment of that edge effectively cuts apart the two parts of the surface of the torus near that segment.
And if you actually cut the surface of the torus all the way along that edge, you end up with a strip with two twists.
The boundary of that surface is two circles.
So no, with the constraints that you are not allowed to actually occupy or color the points of a true topological Möbius strip, then you cannot distinguish it from a cylinder using only local observations.
A: 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.
A: Yes. When you return to the position of where you started, your right hand will be where your left hand was. Technically, this doesn't prove you're on a Mobius strip, only that you're on a non-orientable surface, but it's a step in the right direction.
A: As a side note, there was an old Soviet SF novel about people living in a clearly artificial environment, where a social experiment is conducted. The place they live in is not necessarily a Möbius strip, but has some strange topological properties, such as there is a cliff on the one side and abyss on the other. People, who crash or are thrown down the abyss, "loop" and appear to have fallen from the cliff. There are some more, but that's a major spoiler.
And no, they did not notice.
Well, some did – of else we would not know it, as the story follows a protagonist, – but after an elaborate, arcane, and possibly out-of-that-world research which involved experiencing the non-mentioned further metric issues. The general populace never notices and did not bother.
A: I think that if there are two persons walking on a cylinder, one in front and one behind it at some distance, the person from back sees the person in front vertical, whereas on a Moebius strip, the person in back would see, at some point, the person in front ... off-vertical ?!
A: Assuming the person is held to the Mobius Strip or cylinder through a central gravitational force, yes. 
That person would begin to feel differences as they walked. Since the Mobius Strip has only one side, and the gravitational direction is towards the center of the strip, the person would eventually feel like they were walking on a slope. Eventually, they'd need to climb along the walls, then ceiling, and so on.
A: The boundary of a cylinder is homeomorphic to two circles $S^1 \sqcup S^1$ and the boundary of a mobius strip is homeomorphic to one circle $S^1$, i.e. a mobius strip only has one edge.
A: Get a can of yellow paint, punch a hole in the bottom, and start walking along the boundary.  Assuming the traveler can measure angle, stop after 360 degrees.  Did your yellow paint drips complete the circle?  Or, do you see your paint trail across the road?  
If you see it across the road, keep walking.  After another 360 degrees, did your paint trail close the loop?
A: For this purpose, I will assume that the 'strip' is seemingly straight walkway, with a barrier on either side, so you can't peer over the edge, and a fog or mist to prevent you seeing more than a couple of meters ahead or behind.  'Gravity' or and analogue always points towards the surface of the strip.  (Alternatively, the structure is in space and you are attached to a metal surface via magnets)
Draw a line down the center of the 'strip' you are on, until you reach the start again.  Then, drill a hole on the Right side of line through the entire depth of the strip, and follow the line until you find a hole.
If it is on the Right side of the line, you are on a cylinder.  If it is on the Left then you are on a Mobius Strip.
The important aspect is that you require some form of indication that can be identified from both sides of the strip/surface
A: To determine whether you are on a möbius strip you would need to change your position in a direction orthogonal to the shortest distance between the edges at your sides at the moment of measuring at your current location. If you are on a möbius strip, you will return to your original position after a traversal of 360 degrees times the number of turns in mobius where the number of turns is an odd number.
A: Get surveying tools.
Mark your starting location.
Using them, work out your the relative position in 3-space as you move along the path.
If you first return to your starting location (or right next to it) without being able to see the mark, then next return to it and see the mark, you are on a mobius strip or similar non-orientable surface.
You can work out the exact geometry by extremely careful surveying the shape.
If you cannot get surveying tools, try finding a geometer.
A: I don't think it's the same as walking on a cylinder. On the strip you should see at least 3 different stages: 1. walking as on the cylinder (i.e. not seeing too far away - low horizon). 2. turn to right 3. Go up (like being on the bottom of a hill). Repeat. 
So it should be possible by measuring the vector of the forces if forces are available. In this case it's easy.
A: I assume the Möbius band is a compact surface with boundary $(M, \partial M)$.  The question is a local to global question. Triangulate the surface, record all the incidences and use this information to compute the relative homology $H_2(M, \partial M)$ with real coefficients. It should be zero telling you that you live on a non-orientable surface. By computing $H_1(M)$ you can then detect that this non-orientable surface is a Möbius band.
