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Source. Section 3.5, printed p. 453 (published PDF).

Statement. For matchings $M,N$ of a finite loopless graph, every component containing an edge in the subgraph of $M\triangle N$ is an alternating path or an even alternating circuit. Each path endpoint is exposed for one of the two matchings.

Proof. At a vertex, at most one incident edge belongs to $M$ and at most one to $N$. If two edges of the symmetric difference meet there, one is in $M\setminus N$ and the other in $N\setminus M$. Thus every edge-containing component has maximum degree two: starting at a degree-one vertex traces a simple path, and a component with all degrees two is a circuit. The colors $M,N$ alternate, so a circuit has even length.

If a path ends at $v$ with an edge of $M\setminus N$, then $v$ has no other $M$ edge. Were an $N$ edge incident to $v$, it could not belong to $M$, and would give a second symmetric-difference edge at the endpoint. Hence $v$ is exposed for $N$. The other case is symmetric. Common edges do not occur in these components. $\square$