Orthogonality of linear (alinear) quasigroups and their parastrophes
  Necessary and sufficient conditions of orthogonality of left (right) linear (alinear) quasigroups in various combinations are given. As corollary we obtain conditions of parastroph orthogonality of left (right) linear (alinear) quasigroups. Any linear (alinear) quasigroup over the symmetric group S_n ($n\neq 2; 6$) is not orthogonal to its (12)-parastrophe. 
V.A. Shcherbacov
math.GR