We prove that the essential range of the gradient of planar Lipschitz maps has a connected rank-one convex hull. As a corollary, in combination with the results in [Faraco, D., and Székelyhidi, Jr., L., Tartar's conjecture and localization of the quasiconvex hull in R2x2, Acta Math., to appear.] we obtain a complete characterization of incompatible sets of gradients for planar maps in terms of rank-one convexit
