  New covariant theories of emergent modified gravity exist not only in
spherically symmetric models, as previously found, but also in polarized Gowdy
systems that have a local propagating degree of freedom. Several explicit
versions are derived here, depending on various modification functions. These
models do not have instabilities from higher time derivatives, and a large
subset is compatible with gravitational waves and minimally coupled massless
matter fields travelling at the same speed. Interpreted as models of loop
quantum gravity, covariant Hamiltonian constraints derived from the covariance
conditions found in polarized Gowdy systems are more restricted than those in
spherical symmetry, requiring new forms of holonomy modifications with an
anisotropy dependence that has not been considered before. Assuming homogeneous
space, the models provide access to the full anisotropy parameters of modified
Bianchi I dynamics, in which case different fates of the classical singularity
are realized depending on the specific class of modifications.
