We study a deterministic version of the cellular automaton first shown to display self-organized criticality. Detailed analysis shows that there exist many coexisting periodic attractors, with a period that is independent of initial condition. This leads us to picture the critical state as the union of many such coexisting, neutrally stable orbits. Dhar's recently developed formalism [Phys. Rev. Lett. 64, 1613 (1990)] can be used to explain many of the observed regularities.