  Besides expanding anisotropically, the universe can also be anisotropic at
the level of its (spatial) curvature. In particular, models with anisotropic
curvature and isotropic expansion leads both to a $\Lambda$CDM-like
phenomenology and to an isotropic and homogeneous CMB at the background level.
Thus, they offer an interesting and viable example where the cosmological
principle does not follow from the isotropy of observational data. In this
paper we extract the linear dynamics of tensor perturbations in two classes of
cosmologies with anisotropic spatial curvature. Two difficulties arise in
comparison to the same computation in isotropic cosmologies. First, the two
tensor polarizations do not behave as a spin-2 field, but rather as the spin-0
and spin-1 irreducible components of a symmetric, traceless and transverse
tensor field, each with its own dynamics. Second, because metric perturbations
are algebraically coupled, one cannot ignore scalar and vector modes and focus
just on tensors --- even if one is only interested in the latter --- under the
penalty of obtaining the wrong equations of motion. We illustrate our results
by finding analytical solutions and evaluating the power-spectra of tensor
polarizations in a radiation dominated universe. We conclude with some comments
on how these models could be constrained with future experiments on CMB
polarization.
