I am interested in the ergodic (invertible) transformations $T$ such that $T\times R_\theta$ is ergodic where $R_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all $R_\theta$, only this one).

This includes all weakly mixing transformations, because of the two following results which can be found in Rudolph's book:

If $T$ is ergodic and has no factor isomorphic to an isometry on a compact space, then $T \times S$ is ergodic for every ergodic transformation $S$.

and

If $T$ is weakly mixing then it has no factor isomorphic to an isometry on a compact space.

There are other suitable $T$: every rotation $R_\alpha$ fulfills the desideratum when $\alpha$ and $\theta$ are rationally independent.

Can we expect a characterization of the suitable $T$ ? Or say something more ? Rotations $R_\alpha$ are not suitable when $\alpha$ and $\theta$ are not rationally independent. Are there other transformations which are not suitable ?