  In this paper we investigate a correspondence among spin and vertex models
with the same number of local states on the square lattice with toroidal
boundary conditions. We argue that the partition functions of an arbitrary
$n$-state spin model and of a certain specific $n$-state vertex model coincide
for finite lattice sizes. The equivalent vertex model has $n^3$ non-null
Boltzmann weights and their relationship with the edge weights of the spin
model is explicitly presented. In particular, the Ising model in a magnetic
field is mapped to an eight-vertex model whose weights configurations combine
both even and odd number of incoming and outcoming arrows at a vertex. We have
studied the Yang-Baxter algebra for such mixed eight-vertex model when the
weights are invariant under arrows reversing. We find that while the Lax
operator lie on the same elliptic curve of the even eight-vertex model the
respective $\mathrm{R}$-matrix can not be presented in terms of the difference
of two rapidities. We also argue that the spin-vertex equivalence may be used
to imbed an integrable spin model in the realm of the quantum inverse
scattering framework. As an example, we show how to determine the
$\mathrm{R}$-matrix of the 27-vertex model equivalent to a three-state spin
model devised by Fateev and Zamolodchikov.
