We present ab initio and k·p calculations of the spin texture on the Fermi surface of tensile-strained HgTe, which is obtained by stretching the zinc-blende lattice along the (111) axis. Tensile-strained HgTe is a semimetal with pointlike accidental degeneracies between a mirror symmetry protected twofold degenerate band and two nondegenerate bands near the Fermi level. The Fermi surface consists of two ellipsoids which contact at the point where the Fermi level crosses the twofold degenerate band along the (111) axis. However, the spin texture of occupied states indicates that neither ellipsoid carries a compensating Chern number. Consequently, the spin texture is locked in the plane perpendicular to the (111) axis, exhibits a nonzero winding number in that plane, and changes winding number from one end of the Fermi ellipsoids to the other. The change in the winding of the spin texture suggests the existence of singular points. An ordered alloy of HgTe with ZnTe has the same effect as stretching the zinc-blende lattice in the (111) direction. We present ab initio calculations of ordered HgxZn1−xTe that confirm the existence of a spin texture locked in a 2D plane on the Fermi surface with different winding numbers on either end.