This paper considers Latin squares of order having down the main diagonal and in which the back diagonal is a permutation of these symbols (diagonal squares). It is an open question whether or not such a square which is self-orthogonal (i.e., orthogonal to its transpose) exists for order . We consider two possible constraints on the general concept: self-conjugate squares and strongly symmetric squares. We show that relative to each of these constraints, a corresponding self-orthogonal diagonal Latin square of order does not exist. However, it is easy to construct self-orthogonal diagonal Latin squares of orders and which satisfy each of the constraints respectively.
