Asymmetric self-excited periodic motions or periodic solutions which are produced by relay feedback systems that have symmetric characteristics are studied in the paper. Two different mechanisms of producing an asymmetric oscillation by a system with symmetric properties are noted and analyzed by the locus of a perturbed relay system (LPRS) method. Bifurcation between the ability to excite symmetric and asymmetric oscillation with variation of system parameters is analyzed. An algorithm of finding asymmetric solutions is proposed.