  The $\nu=0$ quantum Hall state in a defect-free graphene sample is studied
within the framework of quantum Hall ferromagnetism. We perform a systematic
analysis of the pseudospin anisotropies, which arise from the valley and
sublattice asymmetric short-range electron-electron (e-e) and electron-phonon
(e-ph) interactions. The phase diagram, obtained in the presence of generic
pseudospin anisotropy and the Zeeman effect, consists of four phases
characterized by the following orders: spin-polarized ferromagnetic, canted
antiferromagnetic, charge density wave, and Kekul\'{e} distortion. We take into
account the Landau level mixing effects and show that they result in the key
renormalizations of parameters. First, the absolute values of the anisotropy
energies become greatly enhanced and can significantly exceed the Zeeman
energy. Second, the signs of the anisotropy energies due to e-e interactions
can change upon renormalization. A crucial consequence of the latter is that
the short-range e-e interactions alone could favor any state on the phase
diagram, depending on the details of interactions at the lattice scale. On the
other hand, the leading e-ph interactions always favor the Kekul\'{e}
distortion order. The possibility of inducing phase transitions by tilting the
magnetic field is discussed.
