For every positive integer n greater than 4 there is a set of Latin squares of order n such that every permutation of the numbers 1, . . . , n appears exactly once as a row, a column, a reverse row or a reverse column of one of the given Latin squares. If n is greater than 4 and not of the form p or 2p for some prime number p congruent to 3 modulo 4, then there always exists a Latin square of order n in which the rows, columns, reverse rows and reverse columns are all distinct permutations of 1, . . . , n, and which constitute a permutation group of order 4n. If n is prime congruent to 1 modulo 4, then a set of (n − 1) / 4 mutually orthogonal Latin squares of order n can also be constructed by a classical method of linear algebra in such a way, that the rows, columns, reverse rows and reverse columns are all distinct and constitute a permutation group of order n (n − 1).