A symmetric Latin square is one that is equal to its matrix transpose. That means that the set of (row,column,symbol) triples is unchanged by swapping the first two coordinates.

We start with catalogues of all reduced symmetric Latin squares:

Then symmetric Latin squares up to isomorphism: ...and the rrs-isotopism classes of symmetric Latin squares... ...and the species that contain symmetric Latin squares:- order 8
For other orders up to 9, just use the rrs-isotopism class representatives above.

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