It has been a long-standing puzzle why clouds, which should interact with solar radiation similarly to a thin layer of snow, have such a dramatic effect on the reflectance observed by satellites over snow-covered regions. The presence of a cloud over the snow strongly enhances the anisotropy of the scene, so that a cloud-over-snow scene appears darker than clear sky over snow when observed near nadir, but much brighter when observed at large zenith angles in the forward reflected direction. By contrast, when a plane-parallel cloud is placed above a plane-parallel snow surface in a model, it slightly decreases the anisotropy of the system because of the cloud's smaller particles. Using a parameterization for the directional reflectance from East Antarctic snow, developed from extensive near-surface observations from a tower, we show that the unexpected effect of clouds over snow in this region is due to the non-plane-parallel nature of the snow surface, not to unexpected features of the clouds. The snow surface roughness reduces the anisotropy of the reflected sunlight compared to that from a plane-parallel snow surface. Clouds, by hiding this roughness with a surface that is very smooth in units of optical depth, increase the anisotropy by bringing the system closer to the plane-parallel case. We use the surface parameterization to accurately model reflectance observations made from the tower over a ground fog and from the top of the atmosphere over cloud-covered snow by the MISR satellite instrument.