Let $X$ and $Y$ be finite sets. I am interested in subsets $r \subseteq X \times Y$, which contain each $x \in X$ and each $y \in Y$ at least once: $$ \forall_{x \in X} \exists_{y \in Y} (x, y) \in r \land \forall_{y \in Y} \exists_{x \in X} (x, y) \in r . $$ For example, one such subset is the whole product itself. In other words, I'm talking about "left-total" and "right-total" relations.
My question: Are such relations known under other names?
In universal algebra, a subalgebra of a direct product with this property is called "subdirect".
By the way, this Wikipedia article validates your terminology.