Some point in the scrumpled version is exactly above its original position.
Another way of illustrating it is that if you're in city X and place a map of city X on the ground, then exactly one point on the map will be above the corresponding physical point.
For the aforesaid property any form of crumpling that does not tear the paper would suffice. Provided no part of the crumpled paper extends beyond the boundary of the pristine sheet (or the city, in your example). The phenomenon relies of Browder's fixed point theorem.
If it helps to reduce one dimension: think of a continuous curve defined over a part of the x-axis [0, 10]. As long as the curve stays inside the box [0,0], [10,10] and every point in the [0,10] part of the x-axis is mapped, it would be impossible to avoid the diagonal. Just try it.
Incidentally, a generalization of the theorem above, called the Kakutani fixed point theorem underlies John Nash's proof (that won him the Nobel) of the existence of equilibria in games.