Integrated photonics is a remarkable platform for scalable classical and quantum light-based information processing. However, polarization manipulation on a chip despite of its fundamental significance in information processing remains elusive. Polarization manipulation capabilities have been recently demonstrated in femtosecond laser-inscribed twisted waveguides, although the systematic theoretical description of polarization manipulation has not been established for this architecture. In this paper we develop a rigorous theory of a twisted waveguide unveiling its eigenmodes and transmission matrix in the closed form. Utilizing the developed theory, we demonstrate that twisted waveguides can realize virtually arbitrary polarization transformations while satisfying reasonable design constraints. This fact combined with low cost and ease of prototyping of laser inscribed photonic integrated circuits allows us to suggest twisted waveguide as a robust building block for on-chip polarization-encoded information processing.