  Binary relations derived from labeled rooted trees play an import role in
mathematical biology as formal models of evolutionary relationships. The
(symmetrized) Fitch relation formalizes xenology as the pairs of genes
separated by at least one horizontal transfer event. As a natural
generalization, we consider symmetrized Fitch maps, that is, symmetric maps
$\varepsilon$ that assign a subset of colors to each pair of vertices in $X$
and that can be explained by a tree $T$ with edges that are labeled with
subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$
if and only if $m$ appears in a label along the unique path between $x$ and $y$
in $T$. We first give an alternative characterization of the monochromatic case
and then give a characterization of symmetrized Fitch maps in terms of
compatibility of a certain set of quartets. We show that recognition of
symmetrized Fitch maps is NP-complete. In the restricted case where
$|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps
coincide with class of monochromatic Fitch maps whose graph-representations
form precisely the class of complete multi-partite graphs.
