Consider a spin-1/2 particle. Rotate it about an axis by a 2 pi angle. You are not getting it back into the same state but with sign flipped. If you rotate it further by another 2 pi angle, you will see the particle in the original state. This is a bizarre object, which is beyond our intuition !
If you ask physicists, why is this? The answer is simple : it is an SU(2) physics, not an SO(3) ! If you want to visualize it, please look at Feynman's description of the rotation of a coffee cup in the following links:
Your views on this matter are highly welcome !
