A transmission-type cross-polarization converter is designed based on a frequency selective surface (FSS) and is investigated theoretically and experimentally. The FSS structure is built up with rotationally asymmetric fourfold supercell. The subcell of each supercell is a monopole-via-monopole (MVM) module, which consists of a receiving monopole, a transmitting monopole, and a quasi-wave-guiding structure (a through-via hole passing through the perforated shielding ground) to connect the monopole pair. In each supercell, the receiving and transmitting monopoles are orthogonally oriented to form a chiral geometry. The cross-polarization converter allows an incident wave linearly polarized at arbitrary azimuth to pass through and convert it into its cross-polarization state, while it operates as an FSS filter (without polarization state conversion) for the incidence of both circularly polarized waves. It exhibits a 3.0-dB transmission bandwidth ranging from 5.5 to 6.18 GHz (12%) with the peak insertion loss less than 0.16 dB at 5.8 GHz. The novelties of the reported cross-polarization converter lie in the insensitivity to the incident linear polarization azimuth and very low insertion loss within the investigated frequency band. To verify the proposed concept, a prototype is designed, fabricated, and measured, and good agreement between the simulated and experimental results is observed.