A separation result for countable unions of Borel rectangles

We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${bfSigma}^0_1 !times! {bfSigma}^0_xi$ sets, or by a ${bfPi}^0_1 !times! {bfPi}^0_xi$ set.