On orthogonality of binary operations and squares

Orthogonality of a pair of binary groupoids, left quasigroups and quasigroups from some points of view is studied. Necessary and sufficient conditions of orthogonality of a finite quasigroup and any its parastrophe (conjugate quasigroup in other terminology), including ones in language of quasi-identities, are given. New concept of gisotopy, which generalizes the concept of isotopy, is defined. There is information on quasigroups with self-orthogonal conjugates.