Let $G$ be a finite graph, and let $D,A,C$ be the Gallai-Edmonds sets. Then every component of $G[D]$ is factor-critical, $G[C]$ has a perfect matching, and every maximum matching of $G$ consists of a perfect matching of $G[C]$, near-perfect matchings in the components of $G[D]$, and a matching that saturates $A$ by assigning each vertex of $A$ to a distinct component of $G[D]$ and using one edge from that vertex into its assigned component. Moreover, the unmatched vertices in any maximum matching lie in components of $G[D]$ not matched to $A$.