Take another copy of SO(3) also looking like the inside of a ball, and glue that onto our original along their spherical boundaries. Leaving a ball turns you over, but reentering turns you upright again!
No matter how you cross Spin(3) you never get back to your starting point turned upside down. The twist is gone! This pair of hemiballs is called 3-sphere or S³. Psst, this object actually lives in #4d! Now you know a bit more about how that works!
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