  We study integrable vertex models and quantum spin chains with toroidal
boundary conditions. An interesting class of such boundaries is associated with
non-diagonal twist matrices. For such models there are no trivial reference
states upon which a Bethe ansatz calculation can be constructed, in contrast to
the well-known case of periodic boundary conditions. In this paper we show how
the transfer matrix eigenvalue expression for the spin-$s$ XXZ chain twisted by
the charge-conjugation matrix can in fact be obtained. The technique used is
the generalization to spin-$s$ of the functional relation method based on
``pair-propagation through a vertex''. The Bethe ansatz-type equations obtained
reduce, in the case of lattice size $N=1$, to those recently found for the
Hofstadter problem of Bloch electrons on a square lattice in a magnetic field.
