  We describe the fine structure of Dirac states in HgTe/CdHgTe quantum wells
of critical and close-to-critical thickness and demonstrate the formation of an
anticrossing gap between the tips of the Dirac cones driven by interface
inversion asymmetry. By combining symmetry analysis, atomistic calculations,
and k-p theory with interface terms, we obtain a quantitative description of
the energy spectrum and extract the interface mixing coefficient. The
zero-magnetic-field splitting of Dirac cones can be experimentally revealed in
studying magnetotransport phenomena, cyclotron resonance, Raman scattering, or
THz radiation absorption.
