  Bilayer graphene (BLG) offers a rich platform for broken symmetry states
stabilized by interactions. In this work we study the phase diagram of BLG in
the quantum Hall regime at filling factor $\nu=0$ within the Hartree-Fock
approximation. In the simplest non-interacting situation this system has eight
(nearly) degenerate Landau levels near the Fermi energy, characterized by spin,
valley, and orbital quantum numbers. We incorporate in our study two effects
not previously considered: (i) the nonperturbative effect of trigonal warping
in the single-particle Hamiltonian, and (ii) short-range SU(4)
symmetry-breaking interactions that distinguish the energetics of the orbitals.
We find within this model a rich set of phases, including ferromagnetic,
layer-polarized, canted antiferromagnetic, Kekul\'e, a "spin-valley entangled"
state, and a "broken U(1) $\times$ U(1)" phase. This last state involves
independent spontaneous symmetry breaking in the layer and valley degrees of
freedom, and has not been previously identified. We present phase diagrams as a
function of interlayer bias $D$ and perpendicular magnetic field $B_{\perp}$
for various interaction and Zeeman couplings, and discuss which are likely to
be relevant to BLG in recent measurements. Experimental properties of the
various phases and transitions among them are also discussed.
