A little more exotic, but if you look here https://en.m.wikipedia.org/wiki/Plate_trick https://en.m.wikipedia.org/wiki/Plate_trick you can read about some sim

A little more exotic, but if you look here https://en.m.wikipedia.org/wiki/Plate_trick https://en.m.wikipedia.org/wiki/Plate_trick you can read about some simple rotating systems that, in some sense, end up reversed after a 360 degree rotation amd require another 360 to turn fully around. The article above has a nifty gif of this. Formally, this kind of thing is studied using what are called spinors and gets used a lot in quantum mechanics and friends when talking about quantum spin. Behind the scenes these are described using things called spinors. As it turns out, quaternions (and octonions) are capable of describing such things while simple rotations of any kind are insufficient. There are deep connections between spinors of different dimensions and the quaternions/octonions. For those interested, the term here is "Bott periodicity".