  We argue that at finite carrier density and large displacement fields,
bilayer graphene is prone to $\ell =0$ and $\ell = 1$ Pomeranchuk Fermi surface
instabilities. The broken symmetries are driven by non-local exchange
interactions which favor momentum space condensation. We find that
electron-electron interactions lead first to spontaneous valley polarization,
which breaks time-reversal invariance and is associated with spontaneous
orbital magnetism, and then under some circumstances to a nematic phase with
reduced rotational symmetry. When present, nematic order is signaled by reduced
symmetry in the dependence of optical absorption on light polarization.
