  We introduce a probabilistic generalization of the dual
Robinson--Schensted--Knuth correspondence, called $qt$RSK${}^*$, depending on
two parameters $q$ and $t$. This correspondence extends the $q$RS$t$
correspondence, recently introduced by the authors, and allows the first
tableaux-theoretic proof of the dual Cauchy identity for Macdonald polynomials.
By specializing $q$ and $t$, one recovers the row and column insertion version
of the classical dual RSK correspondence as well as of $q$- and
$t$-deformations thereof which are connected to $q$-Whittaker and
Hall--Littlewood polynomials. When restricting to Jack polynomials and
$\{0,1\}$-matrices corresponding to words, we prove that the insertion tableaux
obtained by $qt$RSK${}^*$ are invariant under swapping letters in the input
word. Our approach is based on Fomin's growth diagrams and the notion of
probabilistic bijections.
