Can someone explain why it would be 'hard' to decide on the direction of the normal vector given a point on a non-orientable mesh?

If we are thinking of vector fields on a surface instead of just points on a surface, the harry-ball theorem is pretty neat.

Can someone explain why it would be 'hard' to decide on the direction of the normal vector given a point on a non-orientable mesh?

If we are thinking of vector fields on a surface instead of just points on a surface, the harry-ball theorem is pretty neat.