Theorem. Let $M$ and $N$ be perfect matchings in a graph $G=(V, E)$. Then $\chi^M$ and $\chi^N$ are adjacent vertices of the perfect matching polytope if and only if $M \triangle N$ is a circuit.
Proof. To see necessity, let $\chi^M$ and $\chi^N$ be adjacent. Then $M \triangle N$ is the vertexdisjoint union of circuits $C_1, \ldots, C_k$. If $k=1$ we are done so assume $k \geq 2$. Let $M^{\prime}:=M \triangle C_1$ and $N^{\prime}:=N \triangle C_1$. Then $\frac{1}{2}\left(\chi^M+\chi^N\right)=\frac{1}{2}\left(\chi^{M^{\prime}}+\chi^{N^{\prime}}\right)$. This contradicts the adjacency of $\chi^M$ and $\chi^N$.
To see sufficiency, define a weight function $w: E \rightarrow \mathbb{R}$ by $w_e:=0$ if $e \in M \cup N$ and $w_e:=1$ otherwise. Then $M$ and $N$ are the only two perfect matchings in $G$ of minimum weight. Hence $\chi^M$ and $\chi^N$ are adjacent.
Question: In the proof of neccessity, I wonder why we have $\frac{1}{2}\left(\chi^M+\chi^N\right)=\frac{1}{2}\left(\chi^{M^{\prime}}+\chi^{N^{\prime}}\right)$ is contradict with the adjacency of $\chi^M$ and $\chi^N$.
The necessary and sufficient test for two vertices $x,y$ of a polytope $P$ to be adjacent is that there is a vector $w$ such that the set of points $p \in P$ maximizing the inner product $w^{\mathsf T}p$ is the line segment $[x,y]$. (Equivalently, $x$ and $y$ are the only vertices of $P$ that maximize $w^{\mathsf T}p$.) This is also what is being tested in the proof of sufficiency.
If $\chi^M + \chi^N = \chi^{M'} + \chi^{N'}$, then this cannot happen: for any vector $w$ we could try, if $w^{\mathsf T}\chi^M = w^{\mathsf T}\chi^N = w^*$, then $w^{\mathsf T}\chi^{M'} + w^{\mathsf T}\chi^{N'} = 2w^*$, so we must have either $w^{\mathsf T}\chi^{M'} \ge w^*$ or $w^{\mathsf T}\chi^{N'} \ge w^*$. Either option gives a third vertex of $P$ that maximizes $w^{\mathsf T}p$.