  Whereas the concept of topological band-structures was developed originally
for insulators with a bulk bandgap, it has become increasingly clear that the
prime consequences of a non-trivial topology -- spin-momentum locking of
surface states -- can also be encountered in gapless systems. Concentrating on
the paradigmatic example of mercury chalcogenides HgX (X = Te, Se, S), we show
that the existence of helical semimetals, i.e. semimetals with topological
surface states, critically depends on the presence of crystal inversion
symmetry. An infinitesimally small broken inversion symmetry (BIS) renders the
helical semimetallic state unstable. The BIS is also very important in the
fully gapped regime, renormalizing the surface Dirac cones in an anisotropic
manner. As a consequence the handedness of the Dirac cones can be flipped by a
biaxial stress field.
