The dynamics of a system of two diffusively coupled identical unimodal maps is investigated. The unimodal maps have simple kneading sequences which imply that their orbits are asymptotically periodic with period a power of two. Analogous results are obtained for the coupled system for all coupling parameters. We prove the eventual periodicity of the position of orbits with respect to the diagonal of the square phase space and the asymptotic periodicity for orbits whose coordinates have the same sign. Finally, we give a global condition for the existence of symmetric orbits that are examples of orbits with this asymptotic property.