  We present a method to deform (generically non-abelian) T duals of
two-dimensional $\sigma$ models, which preserves classical integrability. The
deformed models are identified by a linear operator $\omega$ on the dualised
subalgebra, which satisfies the 2-cocycle condition. We prove that the
so-called homogeneous Yang-Baxter deformations are equivalent, via a field
redefinition, to our deformed models when $\omega$ is invertible. We explain
the details for deformations of T duals of Principal Chiral Models, and present
the corresponding generalisation to the case of supercoset models.
