Bivariate probability distribution(s) over unit square, uniform marginals, midpoint is saddlepoint

Question

Construct a bivariate probability distribution--or family of such distributions--over the unit square (corners $(0,0), (0,1), (1,1), (1,0)$) with uniform marginals and having a saddlepoint at $(1/2,1/2)$. Perhaps without loss of generality, let the line from $(0,0)$ to $(1,1)$ have its maximum at $(1/2,1/2)$, and the line from $(1,0)$ to $(0,1)$ have its minimum at $(1/2,1/2).$ Of course, the uniform distribution over the square could be regarded as, in some sense, fulfilling the conditions, but I am interested in constructions of a less trivial nature.

Answer 1

Does $f_{X, Y}(x, y) = 2x+2y-4xy$ fit the bill?