Actually the first answer is not entirely correct. Light has momenta p = h/L; where L is wavelength and h in Planck's constant. [See source.] And, as we all know, the change in momentum dp/dt = F, which is force. So as light strikes the rotating mirror in a vacuum, the direction and, therefore, the momentum of light is changed and a force is created that thrusts the mirror around.
But note...p = h/L depends only on wavelength and that is not a quantum or particle. It is in fact just the wave model for light. So the spinning mirror does not prove the particle nature of light at all. It only proves that light has momentum...relativistic momentum because light has no mass.
I think the photoelectric cell and the so-called Compton Effect are the major proofs of the particle nature of light. We all know that certain materials (usually silicon based) create electricity when struck by light. Turns out the effect can only be explained by light particles, waves would not do. Check this out:
"The [Compton] effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency." [See source.]