The topological surface states of mercury telluride (HgTe) are studied by ab initio calculations assuming different strains and surface terminations. For the Te-terminated surface, a single Dirac cone exists at the Gamma-point. The Dirac point shifts up from the bulk valence bands into the energy gap when the substrate-induced strain increases. At the experimental strain value (0.3%), the Dirac point lies slightly below the bulk valence band maximum. A left-handed spin texture was observed in the upper Dirac cone, similar to that of the Bi2Se3-type topological insulator. For the Hg-terminated surface, three Dirac cones appear at three time-reversal-invariant momenta, excluding the Gamma-point, with non-trivial spin textures. Copyright (C) E...