It took me a while to figure out that a "symmetric channel" does not mean a channel with a symmetric channel matrix. (Rather, "symmetric channel" means that the rows of the matrix are all permutations of each other, and so are the columns.) This is obvious in hindsight because an arbitrary channel can have a different number of input symbols and output symbols, in which case the matrix isn't square and so can't possibly be symmetric.
Is there a word for a channel with a symmetric channel matrix (implying it has the same number of input symbols and output symbols)?
Does this property simplify the computation of channel capacity at all?