We study the light propagation in the twisted anisotropic optical fibers endowed with torsional mechanical stress by obtaining the analytical solution o f the vector wave equation. We show that at certain interplay between fiber parameters optical vortex beams o f topological charge l = 0 ,± 1 ,± 2 ,... become the modes o f the fibers in question. To explain the splitting of the optical vortex propagation constants we introduce the notions of orbital birefringence and optical Zeeman effect. Moreover, we unveil that induced by torsional stress circular birefringence makes the vortex beams with the well-defined orbital angular momentum robust against small perturbations characterized by both constant and spatially varying orientation o f a director. We believe that such fibers can be successfully utilized for the long-range robust transmission of information encoded in the light's orbital degrees o f freedom.

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