On arbitrary polygonal domains OmegasubsetRR2, we construct C1 hierarchical Riesz bases for Sobolev spaces Hs(Omega). In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from sin(2,frac52) to sin(1,frac52). Since the latter range includes s=2, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned