Twice around to return home: A hidden reset button for spins and qubits

The world is filled with rotating objects—gyroscopes, magnetic spins, and more recently, qubits in quantum computers. For example, the atomic nuclei in our bodies precess at megahertz frequencies inside NMR machines. In practice, it is often desirable to return such a rotating system precisely to its starting point. At first glance, this seems impossible: after an elaborate sequence of twists and wobbles, how could one possibly retrace the path back to the origin?