The answer to the following question is probably known since long ago, although unknown to me, since I am not a differential geometer.

Let $X$ and $Y$ be 2-dimensional, smooth manifolds and let $Z$ be an open piece of a hypersurface in $X\times Y$ near a point $(x_0, y_0)$ with the properties that both projections $\pi_X:\ Z \to X$ and $\pi_Y:\ Z \to Y$ have surjective differentials and the projections from the conormal $N^*(Z)$ into $T^*(X)$ and $T^*(Y)$ are local diffeomorphisms. What are the invariants of $Z$ with respect to separate diffeomorphisms in $X$ and $Y$? In particular, how can we decide, for instance in terms of a defining function $F(x_1, x_2, y_1, y_2)$ for $Z$, whether there are coordinate systems in $X$ and $Y$ such that the fibers $\pi_X(\pi_Y^{-1}(y))$ and $\pi_Y(\pi_X^{-1}(x))$ are straight lines (which is the case if $F(x, y) = x_1 y_1 + x_2 + y_2$).