Difference b/w Symmetric and Reciprocal Networks
A reciprocal network does indeed have a a symmetric S-matrix i.e. [S]=[S]^T. 

I think confusion often arises here because reciprocity means that the transmission of a signal between any two ports does not depend on the direction of propagation. For the example of a 2-port network this implies only that S12=S21. It does NOT imply S11=S22, so in that sense the network is not identical looking in at each port. That is a more intuitive use of the word symmetric, which people may often use in conversation, hence the confusion.