  The quantum spin Hall (QSH) state, observed in a zero magnetic field in HgTe
quantum wells, respects the time-reversal symmetry and is distinct from quantum
Hall (QH) states. We show that the QSH state persists in strong quantizing
fields and is identified by counter-propagating (helical) edge channels with
nonlinear dispersion inside the band gap. If the Fermi level is shifted into
the Landau-quantized conduction or valence band, we find a transition between
the QSH and QH regimes. Near the transition the longitudinal conductance of the
helical channels is strongly suppressed due to the combined effect of the
spectrum nonlinearity and enhanced backscattering. It shows a power-law decay
1/B^2N with magnetic field B, determined by the number of backscatterers on the
edge, N. This suggests a rather simple and practical way to probe the quality
of recently realized quasiballistic QSH devices using magnetoresistance
measurements.
