A spin-N field goes back to its original state after a rotation of $360^\circ/N$.
So a spin-1 field takes $360^\circ$, a spin-1/2 field takes $720^\circ$ and a spin-2 field takes $180^\circ$.
But this breaks down for scalar fields (called spin-0 fields). As it seems to suggest you must rotate it infinity times to get back to its original state. Whereas in fact you can rotate it an arbitarily small amount.
Hence shouldn't scalar fields be called "spin-infinity fields"? And a true spin-0 field should not look the same under any amount of rotation.
Why is this logic wrong?