Whereas the concept of topological band structures was developed originally for insulators with a bulk band gap, it has become increasingly clear that the prime consequences of a nontrivial topology - spin-momentum locking of surface states - can also be encountered in gapless systems. We show that point-group symmetries allow for helical semimetals, i.e., semimetals with Dirac-like topological surface states, to exist. The presence of this state, however, critically depends on the presence of crystal inversion symmetry. Using the paradigmatic example of mercury chalcogenides HgX (X = Te, Se, S), we show that an infinitesimally small broken inversion symmetry (BIS) renders the helical semimetallic state unstable. The BIS is also very important in the fully gapped topological insulating 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.