  In this work, the linear and gauge-invariant theory of cosmological
perturbations in a class of anisotropic and shear-free spacetimes is developed.
After constructing an explicit set of complete eigenfunctions in terms of which
perturbations can be expanded, we identify the effective degrees of freedom
during a generic slow-roll inflationary phase. These correspond to the
anisotropic equivalent of the standard Mukhanov-Sasaki variables. The
associated equations of motion present a remarkable resemblance to those found
in perturbed Friedmann-Robertson-Walker spacetimes with curvature, apart from
the spectrum of the Laplacian, which exhibits the characteristic frequencies of
the underlying geometry. In particular, it is found that the perturbations
cannot develop arbitrarily large super-Hubble modes.
