720-degree symmetry
One of the most startling facts about a spin-½ particle is that turning it all the way around, a full three hundred and sixty degrees, does not bring its state back to the way it started. The state instead acquires a minus sign. To return the state exactly to itself you must turn it around twice, a full seven hundred and twenty degrees. A spin-½ object, in this precise sense, needs two full turns rather than one.
This sounds impossible because nothing in everyday experience behaves this way; a coffee cup looks the same after one turn. The resolution is that the minus sign sits on the spinor, the mathematical state, not on anything you can directly see in a single measurement. The probabilities of outcomes depend on squared magnitudes, which ignore the sign, so a lone spin gives no obvious clue that anything odd has happened.
Yet the sign is genuinely real, and it can be exposed. In neutron interferometry a beam is split, one half is rotated a full turn in a magnetic field, and when the halves are recombined they interfere as opposites, exactly as the extra minus sign demands. Restoring the original interference takes a second full turn. This double-turn behaviour is a hallmark of half-integer spin and a beautiful demonstration that quantum states obey a richer geometry than ordinary objects.
A single full turn flips the sign of a spin-½ state; only a double turn restores it.
The sign change is not measurable for an isolated spin, since global phase is unobservable. It becomes physical only in interference, where one path's sign can be compared against another's.