Symmetry is at the heart of coding theory. Codes with symmetry, especially

cyclic codes, play an essential role in both theory and practical applications

of classical error-correcting codes. Here we examine symmetry properties for

codeword stabilized (CWS) quantum codes, which is the most general framework

for constructing quantum error-correcting codes known to date. A CWS code Q can

be represented by a self-dual additive code S and a classical code C, i.,e.,

Q=(S,C), however this representation is in general not unique. We show that for

any CWS code Q with certain permutation symmetry, one can always find a

self-dual additive code S with the same permutation symmetry as Q such that

Q=(S,C). As many good CWS codes have been found by starting from a chosen S,

this ensures that when trying to find CWS codes with certain permutation

symmetry, the choice of S with the same symmetry will suffice. A key step for

this result is a new canonical representation for CWS codes, which is given in

terms of a unique decomposition as union stabilizer codes. For CWS codes, so

far mainly the standard form (G,C) has been considered, where G is a graph

state. We analyze the symmetry of the corresponding graph of G, which in

general cannot possess the same permutation symmetry as Q. We show that it is

indeed the case for the toric code on a square lattice with translational

symmetry, even if its encoding graph can be chosen to be translational

invariant.