We study the spin texture of a generic helical liquid, the edge modes of a two-dimensional topological insulator with broken axial spin symmetry. By considering honeycomb and square-lattice realizations of topological insulators, we show that in all cases the generic behavior of a momentum-dependent rotation of the spin quantization axis is realized. Here we establish this mechanism also for disk geometries with continuous rotational symmetry. Finally, we demonstrate that the rotation of spin-quantization axis remains intact for arbitrary geometries, i.e., in the absence of any continuous symmetry. We also calculate the dependence of this rotation on the model and material parameters. Finally, we propose a spectroscopy measurement which should directly reveal the rotation of the spin-quantization axis of the helical edge states.