We present a left-right symmetry restoring method, which removes the detrimental birefringence in the single-mode fiber Sagnac interferometer, achieved with the aid of a half waveplate oriented at a specific angle. We show theoretically and demonstrate experimentally that adding a π-shift between clockwise and counterclockwise propagating, horizontally (in fiber loop plane) polarized field components, the Sagnac loop mirror's reflection becomes independent on birefringence of an element placed in the loop.