Imagine an object moving across a square and when it reaches an edge, it is transported to the opposite side of the square.
Now imagine the same thing, but with a disc. When you reach the edge, you are transported to the opposite side of the disc. In this case, the object gets flipped.
To prevent this, we could divide perimeter into many segments and transport each one without flipping, but this creates discontinuities at the segment borders.
How can we avoid flipping in a continuous way?
I have a vague idea of a solution involving hyperbolic geometry where the perimeter is divided into small segments, but as you approach those segments they get arbitrarily large.
Sorry if this is not very clear. If someone has a clearer way of explaining it, that is welcome. The motivation for this question was thinking about the topology of the universe.
How about we glue a disk to the disk itself! So, the total space would be homeomorphic to $S^2$. Moreover, within the original disk, for each point on the boundary there's a path to the antipode that doesn't flip you around. Seems like the most light-weight no-nonsense solution. I don't think $S^3$ would be an unreasonable guess at the shape of the universe, but if I recall the consensus so far is that our observable universe is pretty flat. That's outside of my comfort zone, though, so that's all I'll say about that.