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!