We analyze the distributions of interplanar angles between interacting side chains with well-defined planar regions, to see whether these distributions correspond to random packing or alternatively show orientational preferences. We use a non-homologous set of 79 high-resolution protein chain structures to show that the observed distributions are significantly different from the sinusoidal one expected for random packing. Overall, we see a relative excess of small angles and a paucity of large interplanar angles; the difference between the expected and observed distributions can be described as a shift of 5% of the interplanar angles from large (> or = 60 degrees) to small (< 30 degrees) values. By grouping the residue pairs into categories based on chemical similarity, we find that some categories have very non-sinusoidal interplanar angle distributions, whereas other categories have distributions that are close to sinusoidal. For a few categories, observed deviations from a sinusoidal distribution can be explained by the electrostatic anisotropy of the isolated pair potential energy. In other cases, the observed distributions reflect the longer range effects of different possible interaction geometries. In particular, geometries that disrupt external hydrogen bonding are disfavored.