  This is the fourth paper in a series that attempt to put forward a consistent
framework for modelling solid regions in neutron stars. Here we turn our
attention to axial perturbations of spherically symmetric spacetimes using a
gauge invariant approach due to one of us. Using the formalism developed in the
first paper in the series it turns out that the matter perturbations are neatly
expressible in terms of a ``metric'' tensor field depending only on the speeds
of shear wave propagation along the principal directions in the solid. The
results are applicable to a wide class of elastic materials and does not assume
material isotropy nor quasi-Hookean behaviour. The perturbation equations are
then specialised to a static background and are given by two coupled wave
equations. Our formalism is thus slightly simpler than the previously existing
results of Shumaker & Thorne, where an additional initial value equation needs
to be solved. The simplification is mainly due to the gauge invariance of our
approach and shows up also in somewhat simpler boundary conditions. We also
give a first order formulation suitable for numerical integration of the
quasi-normal mode problem of a neutron star. The relations between the gauge
independent variables and the, in general, gauge dependent perturbed metric and
strain tensor are explicitly given.
