  We show through both theoretical arguments and numerical calculations that
graphene discerns an unconventional sequence of quantized Hall conductivity,
when subject to both magnetic fields (B) and strain. The latter produces
time-reversal symmetric pseudo/axial magnetic fields (b). The single-electron
spectrum is composed of two interpenetrating sets of Landau levels (LLs),
located at $\pm \sqrt{2 n |b \pm B|}$, $n=0, 1, 2, \cdots$. For $b>B$, these
two sets of LLs have opposite \emph{chiralities}, resulting in {\em
oscillating} Hall conductivity between 0 and $\mp 2 e^2/h$ in electron and hole
doped system, respectively, as the chemical potential is tuned in the vicinity
of the neutrality point. The electron-electron interactions stabilize various
correlated ground states, e.g., spin-polarized, quantum spin-Hall insulators at
and near the neutrality point, and possibly the anomalous Hall insulating phase
at incommensurate filling $\sim B$. Such broken-symmetry ground states have
similarities as well as significant differences from their counterparts in the
absence of strain. For realistic strength of magnetic fields and interactions,
we present scaling of the interaction-induced gap for various Hall states
within the zeroth Landau level.
