  Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of
$G$ the set of colors appearing on the edges incident with $v$. The palette
index of $G$ is defined as the minimum number of such distinct sets, taken over
all possible edge-colorings of $G$. A graph with a small palette index admits
an edge-coloring which can be locally considered to be almost symmetric, since
few different sets of colors appear around its vertices. Graphs with palette
index $1$ are $r$-regular graphs admitting an $r$-edge-coloring, while regular
graphs with palette index $2$ do not exist. Here, we characterize all graphs
with palette index either $2$ or $3$ in terms of the existence of suitable
decompositions in regular subgraphs. As a corollary, we obtain a complete
characterization of regular graphs with palette index $3$.
