  We perform infrared magneto-spectroscopy of Landau level (LL) transitions in
dual-gated bilayer graphene. At $\nu=4$ when the zeroth LL (octet) is filled,
two resonances are observed indicating the opening of a gap. At $\nu=0$ when
the octet is half-filled, multiple resonances are found to disperse
non-monotonically with increasing displacement field, $D$, perpendicular to the
sheet, showing a phase transition at modest displacement fields to the
layer-polarized state with a gap that opens linearly in $D$. When $D=0$ and
$\nu$ is varied, resonances at $\pm\nu$ show an electron-hole asymmetry with
multiple line splittings as the octet is progressively filled. Broadly these
data show good agreement with predictions from a mean-field Hartree-Fock
calculation, but only by accounting for multiple tight-binding terms in a
four-band model of bilayer graphene that also incorporates valley interaction
anisotropy. Our results are consistent with the presence of a canted
anti-ferromagnet (CAFM) ground state at $\nu=0$, and imply the existence of
intermediate phases in the transition from the CAFM to the layer-polarized
regime.
