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It is well known that circularly polarized electromagnetic waves propagating along the magnetic field in the plasma are strongly absorbed when the wave frequency matches the electron cyclotron frequency. This absorption can be eliminated by adding a weak magnetic undulator, leading to the undulator-induced transparency (UIT) of the plasma. Moreover, the group velocity of the waves in the plasma is strongly reduced, resulting in the extreme compression of the wave energy in the plasma. Compressed waves are polarized along the propagation direction and can be used for synchronous electron or ion acceleration. Numerical simulations reveal yet another interesting property of the electromagnetic waves in UIT plasma: strong coupling and conversion between two circular wave polarizations. Depending on how important this cross-polarization effect is, several propagation regimes have been identified and explored by fluid and particle-in-cell simulations.