Symmetric factorial designs in blocks

We consider factorial designs in blocks, where there are two treatment factors with the same number of levels, and both must be orthogonal to blocks. It is shown that these designs are duals of semi-Latin squares, and that the dual is optimal if and only if the semi-Latin square is optimal, for a wide range of optimality criteria. The optimal designs are described in language relevant for the factorial setting, which is shown to have applications in experiments on the interaction between humans and machines.