  We discuss linear perturbations of the most general class of teleparallel
spacetimes with cosmological symmetry, and perform a decomposition of these
perturbations into irreducible components. We then study their behavior under
gauge transformations, i.e., infinitesimal diffeomorphisms of the background
spacetime, and derive a full set of gauge-invariant perturbation variables.
Comparing our results with perturbations around the diagonal tetrad
corresponding to a spatially flat teleparallel background cosmology, we see the
appearance of new terms which depend on the particular choice of a
cosmologically symmetric background tetrad. Finally, we demonstrate the
application of our findings to teleparallel gravity theories. For convenience,
results are displayed both in general and conformal coordinates.
