  We develop a new perturbative framework for studying the r-modes of rotating
superfluid neutron stars. Our analysis accounts for the centrifugal deformation
of the star, and considers the two-fluid dynamics at linear order in the
perturbed velocities. Our main focus is on a simple model system where the
total density profile is that of an $n=1$ polytrope. We derive a partially
analytic solution for the superfluid analogue of the classical r-mode. This
solution is used to analyse the relevance of the vortex mediated mutual
friction damping, confirming that this dissipation mechanism is unlikely to
suppress the gravitational-wave driven instability in rapidly spinning
superfluid neutron stars. Our calculation of the superfluid r-modes is
significantly simpler than previous approaches, because it decouples the r-mode
from all other inertial modes of the system. This leads to the results being
clearer, but it also means that we cannot comment on the relevance of potential
avoided crossings (and associated "resonances") that may occur for particular
parameter values. Our analysis of the mutual friction damping differs from
previous studies in two important ways. Firstly, we incorporate realistic
pairing gaps which means that the regions of superfluidity in the star's core
vary with temperature. Secondly, we allow the mutual friction parameters to
take the whole range of permissible values rather than focussing on a
particular mechanism. Thus, we consider not only the weak drag regime, but also
the strong drag regime where the fluid dynamics is significantly different.
