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We define two complete sets \(\mathcal{L}\) and \(\mathcal{L}’\) of pairwise orthogonal \(9 \times 9\) Latin squares to be equivalent if and only if \(\mathcal{L}’\) can be obtained from \(\mathcal{L}\) by some combination of: (i) applying a permutation \(\theta\) to the rows of each of the \(8\) squares in \(\mathcal{L}\), (ii) applying a permutation \(\phi\) to the columns of each square from \(\mathcal{L}\), and (iii) permuting the symbols separately within each square from \(\mathcal{L}\).
We use known properties of the projective planes of order \(9\) to show that, under this equivalence relation, there are \(19\) equivalence classes of complete sets. For each equivalence class, we list the species and transformation sets of the \(8\) Latin squares in a complete set. As this information alone is not sufficient for determining the equivalence class of a given complete set, we provide a convenient method for doing this.