The high packing density of residues in proteins ought to be manifested in some order; to date this packing order has not been thoroughly characterized. The packing regularity in proteins is important because the internal organization of proteins can have a dominant effect on functional dynamics, and it can aid in the design, simulation and evaluation of structures. Packing metrics could also inform us about normal sequence variability, an issue that, with the accumulating genome data, becomes increasingly important. Other studies, indicating a possible correlation between packing density, sequence conservation, and folding nucleation [O. B. Ptitsyn, J. Mol. Biol. 278, 655 (1998)], have emphasized the importance of packing. Here, residue clusters from protein databank structures, each comprised of a central residue and all neighbors located within the first coordination shell, have been rigidly re-oriented and superimposed in a self-consistent optimization. About two-thirds of residues are found to follow approximately the relative orientation preferences of face-centered-cubic packing, when examined on a coarse-grained scale (one site per residue), while the remaining one-third occupy random positions. The observed regularity, which becomes more pronounced after optimal superimposition of core residues, appears to be the result of uniform sampling of the coordination space around each residue on a coarse-grained scale with hydrophobic clustering and volume exclusion, to achieve packing densities close to that of the universal closest packing of identical spheres.