We present a construction of C1 piecewise quadratic hierarchical bases of Lagrange type on arbitrary polygonal domains Ω⊂R2. Properly normalized, these bases are Riesz bases for Sobolev spaces Hs(Ω), with [Formula presented]. The method is applicable to arbitrary initial triangulations of polygonal domains, and does not require a checkerboard quadrangulation needed for earlier C1 cubic hierarchical Lagrange bases. Homogeneous boundary conditions can be taken into account in a natural way, and lead to Riesz bases for Sobolev spaces H0s(Ω), s∈(1,5/2)∖3/2, and H003/2(Ω).