I shall explain how any non-degenerate vector field on a bounded domain of
$R^n$ is monotone modulo a measure preserving involution $S$ (i.e., $S^2=Identity$). This is to be compared to Brenier\'s polar decomposition which yields that any such vector field is the gradient of a convex function (i.e., cyclically monotone) modulo a measure preserving transformation.
Connections to mass transport --which is at the heart of Brenier\'s
decomposition-- is elucidated. This is joint work with A. Momeni.