  The inhomogeneous six-vertex model is a 2$D$ multiparametric integrable
statistical system. In the scaling limit it is expected to cover different
classes of critical behaviour which, for the most part, have remained
unexplored. For general values of the parameters and twisted boundary
conditions the model possesses ${\rm U}(1)$ invariance. In this paper we
discuss the restrictions imposed on the parameters for which additional global
symmetries arise that are consistent with the integrable structure. These
include the lattice counterparts of ${\cal C}$, ${\cal P}$ and ${\cal T}$ as
well as translational invariance. The special properties of the lattice system
that possesses an additional ${\cal Z}_r$ invariance are considered. We also
describe the Hermitian structures, which are consistent with the integrable
one. The analysis lays the groundwork for studying the scaling limit of the
inhomogeneous six-vertex model.
