Spinors are double covers for rotation groups. Double covering sounds like it would become more complicated. But on the contrary, it really makes things simpler: Take SO(3): the space of rotations in 3-space! Pick an axis of a sphere: a pair of antipodal points. We can then rotate around that axis. The result is a closed 3-space.

https://en.wikipedia.org/wiki/3D_rotation_group

It looks like a ball where, if you try to leave through the wall, you reappear on the opposite side, turned upside down! Not mirrored, turned.