Enumeration of Latin squares with conjugate symmetry

A Latin square has six conjugate squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a symmetry if at least two of conjugates are equal. enumerate with and classify them according to several common notions equivalence. also do similar enumerations under additional hypotheses, such as assuming the is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature suggested patterns we then found proofs for, including (1) number isomorphism classes semisymmetric order n equals unipotent + 1, (2) suppose B totally symmetric ≢ 0 mod 3. If paratopic isomorphic.