Totally symmetric Latin squares and cubes

The standard definition of a symmetric Latin square, in which the symmetry is between rows and columns, is extended to that of a totally symmetric Latin square, which additionally has analogous symmetries between rows and symbols and between columns and symbols, the rows, columns and symbols all being labelled in the same way, say by 1, 2, ..., n. Details are given of the enumeration of totally symmetric n \times n squares, n \leq 7. Totally symmetric complete Latin squares are discussed. Totally symmetric Latin cubes of size n \times n \times n are defined too, and details of their enumeration are given, n \leq 5.