A bipartite graph $\Gamma$ together with an embedding into a surface $\Sigma$ determines a cluster algebra $\mathcal{A}$ where each four-valent BFZ mutation corresponds to a local $\text{180}^\circ \text{- rotation}$ of a quadrilateral face of $\Gamma$ within the surface. In this framework each choice of boundary condition $I$ and homology class $\gamma \in \text{H}_1\big(\Sigma; \Bbb{F}_2 \big)$ determine a (normalised) dimer partition function $Z_{I,\gamma}$ which is invariant under any four-valent mutation. Conjecturally $Z_{I,\gamma}$ will be a cluster variable in $\mathcal{A}$. A spin-structure on a closed surface $\Sigma$, at least in topological terms, is synonymous with a $\Bbb{F}_2$-valued quadratic form $q$ on the first homology group $\text{H}_1\big(\Sigma; \Bbb{F}_2 \big)$. Following Cimasoni and Reshetikhin, one can construct a spin structure using an embedded (bipartite) graph $\Gamma \hookrightarrow \Sigma$ together with a choice of Kasteleyn orientation and dimer configuration for the graph. I would like to explain BFZ mutation in this context: In particular how a Kasteleyn orientation and a dimer configuration can be transported by four-valent BFZ mutation and how, conjecturally, BFZ mutation must preserve the associated spin-structure.