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We explore the effect of directionality on rotational and translational relaxation in glassy systems of patchy particles. Using molecular dynamics simulations we analyze the impact of two distinct patch geometries, one that enhance local icosahedral structure and one which does not strongly affect local order. We find that in nearly all investigated cases, rotational relaxation takes place on a much faster time scale than translational relaxation. By comparing to a simplified dynamical Monte Carlo model, we illustrate that rotational diffusion can be qualitatively explained as purely local motion within a fixed environment, which is not coupled strongly to the cage-breaking dynamics required for translational relaxation. Nonetheless, icosahedral patch placement has a profound effect on the local structure of the system, resulting in a dramatic slowdown at low temperatures which is strongest at an intermediate "optimal" patch size.