We have computed the rotational diffusion coefficient for a suspension of hard spheres. We find excellent agreement with experimental results over a density range up to, and including, the colloidal crystal. However, we find that theories derived to second order in the volume fraction overestimate the rotational diffusion coefficient for volume fractions exceeding 25%. To investigate the sensitivity of the rotational diffusion coefficient to the pair distribution function we also consider a perfect FCC crystal with negligible thermal motion. We show that, in line with theoretical predictions, the first term in the expansion of the rotational diffusion coefficient in powers of the volume fraction becomes quadratic. Relative to the random distribution, the rotational diffusion coefficient in this case is significantly larger. By studying the decay of angular velocity fluctuations, we examined the time dependence of the rotational diffusion coefficient. We find that for rotation the situation is similar to that reported for translation. The suspension behaves like an "effective fluid", i.e. the rotational dynamics of a particle in the suspension can be described by the isolated particle result, but with the suspension viscosity replacing the fluid viscosity. As with translation, this picture only holds for times long compared to the time it takes transverse momentum to diffuse over a distance of the order of a particle radius.