Multi-dimensional potential energy surfaces are associated with optical binding. A detailed exploration of the available degrees of geometric freedom reveals unexpected turning points, producing intricate patterns of local force and torque. Although optical pair interactions outweigh Casimir-Polder coupling even over short distances, the forces are not always attractive. Numerous local potential minimum and maximum can be located, and mapped on contour diagrams. Islands of stability appear, and structures conducive to the formation of rings. The results, based on quantum electrodynamics, apply to optically trapped molecules, nanoparticles, microparticles and colloids.