A simple theoretical formalism is developed to describe the effect of transmission on linearly polarized light through a fiber with random fluctuations of birefringence. We conclude that, for any optical fiber that does not experience polarization-dependent gain or loss, there exist two orientations for linearly polarized light input into the optical fiber that will also exit the fiber linearly polarized. We report experimental results that verify this prediction and also investigate its practical implications and limitations; in particular we investigate the stability of these linearly polarized output states in laboratory conditions.