We explore the connection between reversibility and shear-induced ordering in non-Brownian suspensions of spheres or rods that are subjected to periodic strain.  We find a number of remarkable results.  We find that the shear-induced random collisions that usually produce diffusive dynamics can also lead to a non-equilibrium phase transition from a fluctuating diffusing state to a self-organized quiescent state (no diffusion).  The reversibility of the system can be probed through time-dependent rheological experiments.  The rheological data exhibit a universal scaling that is associated with a well-known class of non-equilibrium phase transitions associated with directed percolation.  In suspensions of rods, we  find that oscillatory shear flow tends to align rods perpendicular to the flow along the vorticity direction rather than aligning the system along the flow direction as is more commonly observed in shear flows.