This is a road, where people drive on the left side:
Everyone knows exactly what direction to drive in, no matter where on the road they are.
Consider a more 'complicated' shape, like a circular road:
People can still drive on the correct (left) side of the road (locally speaking), no matter where they are.
Now I'm wondering, are there any exotic road shapes (not necessarily physically realizable) where one cannot assign a well-defined driving direction at each point on the road, in a smooth way? (I mean, there'll definitely be a line of discontinuity along the divider where driving direction flips, but apart from that). Let's say that intersections are not allowed.
How would these ideas be formulated mathematically?