How do you design an object that looks different after you spin 360 degrees? According to quantum mechanics, after a 360 rotation electrons have the opposite phase. 
If you rotated yourself 360 degrees, your electrons would have the opposite phase to the electrons in the Universe than they used to have.
Therefor, could you create an object such as a picture that was, for example, red but when you rotated yourself 360 degrees it would appear blue and then after 720 degrees it looked red again.
Therefor using this picture you would know how many rotations you have made.
Perhaps it would involve some special glasses you would have to wear. 
But in principle could such an object be made? It would be interesting to make such an object and then two people might see something different depending on how many rotations they have made!
 A: here is a fun thing to try which may shed some light on your question: 
you can prepare yourself just such a system right now, as follows: place a coffee cup in the palm of your outstretched right hand. while holding the cup level so it does not fall out of your hand, rotate your arm in towards your body so the cup in your palm passes by your ribs on your right side. continue rotating your arm in that direction and raise your arm while doing so. bring your hand around in front of your face this way; in so doing you have rotated the cup in your palm by 360 degrees. now note your arm: it has a twist in it that it did not have before. the system of your arm and the cup is now in a distinctly different state after a 360 degree rotation. 
to revert this system to its original configuration requires you to continue the rotation of your palm with the cup in it through another 360 degrees, which you can do by continuing to revolve your upper arm around your elbow.
try this in stages in a mirror to see how 720 degrees of rotation is needed to get the system back to  where it started. 
A: There is another way to demonstrate the $4\pi$ rotation is indeed the identity while $2\pi$ is not. Namely, if you take a strip and connect it to a fixed place, then rotate the other end by $2\pi$ (you can glue a pencil to indicate the orientation of that end), it will be twisted and there is no way to untwist it by any kind of translation in 3D space. 
However, if you rotate it $2\pi$ again in the same direction, then you can translate the twisted end through itself (like o closed path) and bring it to the original position where you see the twist has been undone.
