  We present magnetotransport measurements of a strongly hybridized inverted
InAs/GaSb double quantum well. We find that the spin-orbit interaction leads to
an appreciable spin-splitting of hole-like states, which form distinct Landau
levels in a perpendicular magnetic field. The resulting quantum Hall state is
governed by a periodic even and odd total filling arising due to the
simultaneous occupation of electron-like and hole-like Landau levels of
differing degeneracy. Furthermore, oscillatory charge transfer between all
involved subbands leads to discrete phase slips in the usual sequential filling
of Landau levels, and coincidentally the phase slips are close to $\pi$. These
results shed new insights on the Landau level structure in composite systems
and have consequences for interpreting intercepts obtained from index plots,
which are routinely employed to investigate the presence of Berry's phase.
