  A Latin square has six conjugate Latin squares obtained by uniformly
permuting its (row, column, symbol) triples. We say that a Latin square has
conjugate symmetry if at least two of its six conjugates are equal. We
enumerate Latin squares with conjugate symmetry and classify them according to
several common notions of equivalence. We also do similar enumerations under
additional hypotheses, such as assuming the Latin square is reduced, diagonal,
idempotent or unipotent.
  Our data corrected an error in earlier literature and suggested several
patterns that we then found proofs for, including (1) The number of isomorphism
classes of semisymmetric idempotent Latin squares of order $n$ equals the
number of isomorphism classes of semisymmetric unipotent Latin squares of order
$n+1$, and (2) Suppose $A$ and $B$ are totally symmetric Latin squares of order
$n\not\equiv0\bmod3$. If $A$ and $B$ are paratopic then $A$ and $B$ are
isomorphic.
