  We study the inhomogeneous 8-vertex model (or equivalently the XYZ Heisenberg
spin-1/2 chain) with all kinds of integrable quasi-periodic boundary
conditions: periodic, $\sigma^x$-twisted, $\sigma^y$-twisted or
$\sigma^z$-twisted. We show that in all these cases but the periodic one with
an even number of sites $\mathsf{N}$, the transfer matrix of the model is
related, by the vertex-IRF transformation, to the transfer matrix of the
dynamical 6-vertex model with antiperiodic boundary conditions, which we have
recently solved by means of Sklyanin's Separation of Variables (SOV) approach.
We show moreover that, in all the twisted cases, the vertex-IRF transformation
is bijective. This allows us to completely characterize, from our previous
results on the antiperiodic dynamical 6-vertex model, the twisted 8-vertex
transfer matrix spectrum (proving that it is simple) and eigenstates. We also
consider the periodic case for $\mathsf{N}$ odd. In this case we can define two
independent vertex-IRF transformations, both not bijective, and by using them
we show that the 8-vertex transfer matrix spectrum is doubly degenerate, and
that it can, as well as the corresponding eigenstates, also be completely
characterized in terms of the spectrum and eigenstates of the dynamical
6-vertex antiperiodic transfer matrix. In all these cases we can adapt to the
8-vertex case the reformulations of the dynamical 6-vertex transfer matrix
spectrum and eigenstates that had been obtained by $T$-$Q$ functional
equations, where the $Q$-functions are elliptic polynomials with
twist-dependent quasi-periods. Such reformulations enables one to characterize
the 8-vertex transfer matrix spectrum by the solutions of some Bethe-type
equations, and to rewrite the corresponding eigenstates as the multiple action
of some operators on a pseudo-vacuum state, in a similar way as in the
algebraic Bethe ansatz framework.
