If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

Question

As in the question title, let $A, B$ be a partition of the unit circle $S^1$, equipped with the Haar measure. Here, we do not require $A, B$ to be measurable. Also, assume neither $A$ nor $B$ is of measure zero, so they are either both non-measurable or both of positive measure. Is it possible, then, for $R_\theta(A) \cap B$ to be measurable and has Haar measure zero for all $\theta$, where $R_\theta$ is the rotation by degree $\theta$?