A semisymmetric Latin square is one that is equal to 3 of its conjugates (also known as parastrophes). That means that the set of (row,column,symbol) triples is unchanged by cyclic permutation of each triple.

We start with catalogues of semisymmetric Latin squares up to isomorphism:

...and the species that contain semisymmetric Latin squares: Next, isomorphism classes of idempotent semisymmetric Latin squares: Next, isomorphism classes of diagonal semisymmetric Latin squares:- order 3
- order 4
- order 5
- order 7
- order 8
- order 9
- order 10
- order 11
(There are none of orders 2 or 6).

If you want to know what format these files are in, it is my usual latin squares format.