  We devote this work to the study of the mean-field phase diagram of the
$\nu=0$ quantum Hall state in bilayer graphene and the computation of the
corresponding neutral collective modes, extending the results of recent works
in the literature. Specifically, we provide a detailed classification of the
complete orbital-valley-spin structure of the collective modes and show that
phase transitions are characterized by singlet modes in orbital pseudospin,
which are independent of the Coulomb strength and suffer strong many-body
corrections from short-range interactions at low momentum. We describe the
symmetry breaking mechanism for phase transitions in terms of the valley-spin
structure of the Goldstone modes. For the remaining phase boundaries, we prove
that the associated exact $SO(5)$ symmetry existing at zero Zeeman energy and
interlayer voltage survives as a weaker mean-field symmetry of the Hartree-Fock
equations. We extend the previous results for bilayer graphene to the monolayer
scenario. Finally, we show that taking into account Landau level mixing through
screening does not modify the physical picture explained above.
