We find that the evolution of an optical polarization along a twisted optical fiber may be spatially synchronized with polarization-selective light scattering. We demonstrate experimentally that linearly polarized light initially oriented along the fast axis of an adiabatically twisted single-mode, polarization-maintaining optical fiber is converted into elliptically and then into circularly polarized light with the same handedness as the chiral structure. As the state of polarization is changing along the length of the fiber, the light is scattered out of the fiber core. By choosing an appropriate twist acceleration profile in custom-made, rectangular-core polarization maintaining fibers, scattering and conversion may be synchronized, allowing the orthogonal polarization to freely propagate through the fiber. When the portion of the fiber with accelerated twist is combined with another fiber segment with decelerated twist, the fiber becomes a broadband, low-insertion-loss, in-fiber, linear polarizer. In this polarizer, the passing component of the incident light oriented along the slow axis, is converted to circularly polarized light of the opposite handedness, and then converted back to linearly polarized light oriented along the same slow axis. While the polarization evolution may be calculated using 1D model of light propagation through a birefringent medium, the calculation of light scattering requires a full 3D calculation.