Let's have two Latin squares, in this case, the two which are shown in this question. If we superimpose them we get 34 different combinations, with two repetitions, namely 4B and 1E. Can someone find two Latin squares, which when superimposed have fewer than two repetitions?
Neither B1 nor E4 appear at all, as a result of B4 and E1 appearing twice. One of the two coordinates for each pair has been permuted with the other pair. Since repeating that permutation would return the original pairs, it is of order 2.
Since the shortest odd permutation that is not identical is order 3, and a permutation identical to a Graeco-Latin square of order 6 is impossible, you cannot make a square that has fewer repeated pairs (or missed pairs) than shown.