Abstract We study the eight-vertex model at its free-fermion point. We express a new “switching” symmetry of the model in several forms: partition functions, order-disorder variables, couplings, Kasteleyn matrices. This symmetry can be used to relate free-fermion 8V-models to free-fermion 6V-models, or bipartite dimers. We also define new solution of the Yang–Baxter equations in a “checkerboard” setting, and a corresponding Z -invariant model. Using the bipartite dimers of Boutillier et al. (Probab Theory Relat Fields 174:235–305, 2019), we give exact local formulas for edge correlations in the Z -invariant free-fermion 8V-model on lozenge graphs, and we deduce the construction of an ergodic Gibbs measure.