A flatlander who lives on a Möbius strip has a 'local' orientation. The stuff that is on his left can, if he travels in a loop that passes through the twist once, be later seen to be on his right, standing at the same spot looking in the same direction.
Consider if you and your dog stick together for all of his training, and you always give him treats with what you perceive as your left hand.
If you then tie up your dog, and go around the strip, when you get back, your dog will expect treats from the wrong hand.
Another way to imagine this is to consider the universe as consisting of two exact copies of you on a cylinder, with one oriented the opposite way. So when you are at $(x,y,z)$ then your duplicate is always at $(-x,-y,-z)$ with $x^2+y^2=1$ and $z\in [-1,1]$. Everything in this universe is similarly duplicated. So, if you travel around to the opposite side and find your "other" dog there, he is oriented differently than your first dog.
You could emulate this in a video game by showing two PacMan on a standard rectangle $[0,1]\times[0,1]$ such that when PacMan 1 is at $(x,y)$ then PacMan 2 is at $(x\pm 1/2,1-y)$. The maze and the ghosts would all have to also be similarly duplicated, and the joystick controls would be mapped from exactly one of the Pac Men - so you'd keep your eye on only one. This is an interesting example, because you'd actually not need to render the "other" you, because if the "other" you is about to die, so are you. So you can actually visualize it entirely as you on a cylinder, with the ghosts having a duplicated behavior, and the maze having a certain inverted symmetry. That might be the best way to render the game for ease of play.
A final way is to give the Möbius strip "depth." This is like my Klein bottle maze, where, when you (a 3D person) get back to a point where you started, you are "flipped" - your feet are on the surface the ceiling used to be on. The Möbius strip in this case is the midpoint between the floor and the ceiling. (There is another way to give the Möbius strip depth, where you just invert orientation like above, but that is essentially the same as before.)
The point of this example is that "left" and "right" invert when you turn upside down. If you face the same direction, but upside down, what was left is now right.
In that case, if you traveled once around the twist, you'd find your dog appeared to be on the ceiling, rather than the floor.