This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related to the $n!$ conjecture, but invariants seem to have become a stepchild in this process.)
Let $k$ be a field of characteristic $0$. Let $n\in\mathbb N$. The group $S_n\times S_n$ acts on the polynomial ring $k\left[X_1,X_2,...,X_n,Y_1,Y_2,...,Y_n\right]$ by