  In this paper we investigate the integrability properties of a two-state
vertex model on the square lattice whose microstates at a vertex has always an
odd number of incoming or outcoming arrows. This model was named odd
eight-vertex model by Wu and Kunz \cite{WK} to distinguish it from the well
known eight-vertex model possessing an even number of arrows orientations at
each vertex. When the energy weights are invariant under arrows inversion we
show that the integrable manifold of the odd eight-vertex model coincides with
that of the even eight-vertex model. The form of the $\mathrm{R}$-matrix for
the odd eight-vertex model is however not the same as that of the respective
Lax operator. Altogether we find that these eight-vertex models give rise to a
generic sheaf of $\mathrm{R}$-matrices satisfying the Yang-Baxter equations
resembling intertwiner relations associated to equidimensional representations.
