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.

Comment: 5 pages. References added. Projector reinstated in eq. (3). Claims on DTD with degenerate cocycle corrected