We perform numerical simulations of athermal repulsive frictionless disks and spheres in two and three spatial dimensions undergoing cyclic quasi-static simple shear to investigate particle-scale reversible motion. We identify three classes of steady
-state dynamics as a function of packing fraction phi and maximum strain amplitude per cycle gamma_{rm max}. Point-reversible states, where particles do not collide and exactly retrace their intra-cycle trajectories, occur at low phi and gamma_{rm max}. Particles in loop-reversible states undergo numerous collisions and execute complex trajectories, but return to their initial positions at the end of each cycle. Loop-reversible dynamics represents a novel form of self-organization that enables reliable preparation of configurations with specified structural and mechanical properties over a broad range of phi from contact percolation to jamming onset at phi_J. For sufficiently large phi and gamma_{rm max}, systems display irreversible dynamics with nonzero self-diffusion.
