In this work, we have demonstrated that self-complementary metasurfaces can provide linear-to-circular polarization conversion. For illustration of this general approach, we studied several particular structures based on periodic self-complementary patterns etched on metal sheets of negligible thickness. The advantage of using self-complementary metasurfaces over previous proposals of thin polarizers is the rigorous constancy of the phase difference between transmission (or reflection) coefficients corresponding to orthogonal linear polarizations, which is just ${90}^{\ensuremath{\circ}}$ at any frequency. Also, this unusual phenomenon is stable under oblique incidence. The general proposed theory of self-complementary metasurfaces was first validated through numerical simulations for structures made of ideal lossless materials. Through simulations with realistic materials and experimental measurements, we have also demonstrated that this conversion can be reached in reality with high precision.