The dynamics of waves in periodic media is determined by the band structure of the underlying periodic Hamiltonian. Symmetries of the Hamiltonian can give rise to novel properties of the band structure. Here we consider a class of periodic Schrödinger operators, HV = −∆ + V , where V is periodic with respect to the lattice of translates Λ = Z 2 . The potential is also assumed to be real-valued, sufficiently regular and such that, with respect to some origin of coordinates, inversion symmetric (even) and invariant under π/2 rotation. 2 with the same notation as in (C).and V 1,0 denote the (1, 1) and (1, 0) indexed Fourier coefficients of V (see (2.4)) and assume the (generically satisfied) non-degeneracy condition: V 1,1 = V 1,0 . Then, for all non-zero and sufficiently small ε, there are 2 dispersion surfaces of H ε , among the lowest 4, that touch at the vertices of B. In a neighborhood of each vertex, the local character given in (1.4) or (1.5).3. Theorem 6: There exists a discrete set C ⊂ R, such that if ε / ∈ C, then 2 dispersion surfaces of H ε touch at the vertices of B with local behavior described by (1.4). The constants α ε ∈ R, β ε ∈ C, and γ ε ∈ C in Theorem 5.3, and α ε and β ε in Theorem 6, displayed in (4.27), depend on the degenerate eigenspace, span{Φ ε 1 , Φ ε 2 }, for quasi-momentum. Hence, the property of quadratically touching dispersion surfaces with local behavior given by (1.5) holds for generic, even arbitrarily large, values of ε.