  There are many concepts of signed graph coloring which are defined by
assigning colors to the vertices of the graphs. These concepts usually differ
in the number of self-inverse colors used. We introduce a unifying concept for
this kind of coloring by assigning elements from symmetric sets to the vertices
of the signed graphs. In the first part of the paper, we study colorings with
elements from symmetric sets where the number of self-inverse elements is
fixed. We prove a Brooks'-type theorem and upper bounds for the corresponding
chromatic numbers in terms of the chromatic number of the underlying graph.
These results are used in the second part where we introduce the
symset-chromatic number $\chi_{sym}(G,\sigma)$ of a signed graph $(G,\sigma)$.
We show that the symset-chromatic number gives the minimum partition of a
signed graph into independent sets and non-bipartite antibalanced subgraphs. In
particular, $\chi_{sym}(G,\sigma) \leq \chi(G)$. In the final section we show
that these colorings can also be formalized as $DP$-colorings.
