  We revisit the problem of inertial r-modes in stratified stars, drawing on a
more precise description of the composition stratification in a mature neutron
star. The results highlight issues with the traditional approach to the
problem, leading us to rethink the computational strategy for r-modes of
non-barotropic neutron stars. We outline two strategies for dealing with the
problem. For moderate to slowly rotating neutron stars the only viable
alternative may be to approach the problem numerically from the outset, while a
meaningful slow-rotation calculation can be carried out for the fastest known
spinning stars (which may be close to being driven unstable by the emission of
gravitational waves). We demonstrate that the latter approach leads to a
problem close, but not identical, to that for barotropic inertial modes. We
also suggest that these reformulations of the problem likely resolve the
long-standing problem of singular behaviour associated with a co-rotation point
in rotating relativistic neutron stars. This issue needs to be resolved in
order to guide future gravitational-wave searches.
