Summary: J.D.H. Smith showed how to replace homotopies between quasigroups by homomorphism between semisymmetric quasigroups. This is a semisymmetrization and it replaces a quasigroup by a semisymmetric structure defined on its Cartesian cube. The reason for a semisymmetrization is that homomorphisms behave more regularly than homotopies. A thorough survey of properties of Smith’s semisymmetrization is given in this paper. Also, new semisymmetrizations, which replace a quasigroup by semisymmetric structures defined on its Cartesian square are suggested.