The goal of this talk is to present new results dealing with the asymptotic joint independence properties of commuting strongly mixing transformations along polynomials. These results form natural strongly mixing counterparts to various weakly and mildly mixing  polynomial ergodic theorems. A decisive role in the proofs is played by a new notion of convergence that is adequate for dealing with strong mixing and, as we will see, cannot be avoided while working with commuting polynomial actions. This talk is based on joint work with Vitaly Bergelson.