  Bilayer graphene, in the presence of a one-sided spin-orbit interaction (SOI)
induced by a suitably chosen substrate, is predicted to exhibit unconventional
Quantum Hall states. The new states arise due to strong SOI-induced splittings
of the eight zeroth Landau levels, which are strongly layer-polarized, residing
fully or partially on one of the two graphene layers. In particular, an Ising
SOI in the meV scale is sufficient to invert the Landau level order between the
$n=0$ and $n=1$ orbital levels under moderately weak magnetic fields $B
\lesssim 10$\~T. Furthermore, when the Ising field opposes the $B$ field, the
order of the spin-polarized levels can also be inverted. We show that, under
these conditions, three different compensated electron-hole phases, with equal
concentrations of electrons and holes, can occur at $\nu = 0$ filling. The
three phases have distinct edge conductivity values. One of the phases is
especially interesting, since its edge conduction can be turned on and off by
switching the sign of the interlayer bias.
