The phase diagram associated with an algebraic model Hamiltonian that encompasses the main physical ingredients of coupled two-dimensional benders is presented. The model is based on a U(3) ? U(3) dynamical algebra and can be applied, for instance, to the study of the bending dynamics of tetratomic molecules. A Hamiltonian with three control parameters, encompassing four dynamical symmetries as limiting cases, is defined and its classical limit is obtained using the intrinsic state formalism. The four possible geometric phases can be associated with linear, cis bent, trans bent, and non-planar configurations. The approach, which is inspired by the dynamics of coupled benders in tetratomic molecular species, is completely general. © 2011 Elsevier B.V. All rights reserved.