The concept of orthogonality of Latin
squares can, of course, be generalized.
The following is from A.E.
Brouwer, Chapter 5, "Recursive Constructions of Mutually Orthogonal Latin
Squares," pp. 149-168 of Latin
Squares: New Developments in the Theory and Applications, J. Denes
and A. D. Keedwell, editors, Annals of Discrete Mathematics, 46,
North-Holland, 1991.
For an earlier (1978)
instance of this generalized concept of orthogonality, see