The Biham-Middleton-Levine traffic model is perhaps the simplest system exhibiting phase transitions and self-organization. Moreover, it is an underpinning to extensive modern studies of traffic flow. The general belief is that the system exhibits a sharp phase transition from freely flowing to fully jammed, as a function of initial density of cars. However, we discover intermediate stable phases, where jams and freely flowing traffic coexist. The geometric structure of such phases is highly regular, with bands of free flowing traffic intersecting at jammed wave fronts that propagate smoothly through the space. Instead of a phase transition as a function of density, we see bifurcation points, where intermediate phases begin coexisting with the more conventionally known phases. We show that the regular geometric structure is in part a consequence of the finite size and aspect ratio of the underlying lattice, and that for certain aspect ratios the asymptotic intermediate phase is on a periodic limit cycle (the exact microscopic configuration recurs each tau time steps). Aside from describing these intermediate states, which previously were overlooked, we derive simple equations to describe the geometric constraints, and predict their asymptotic velocities.