A geometrically-oriented framework for studying the general proton-neutron interacting-boson model of nuclei is presented. The Hamiltonian is resolved exactly into intrinsic (bandhead related) and collective (in-band related) parts. Shape parameters are introduced through non-spherical proton-neutron bases. Genuine intrinsic modes of the combined system, as well as decoupled spurious Goldstone modes, are identified by applying the Bogoliubov treatment to the intrinsic part of the Hamiltonian. The normal frequencies provide an approximate expression for bandhead energies in the limit of large boson numbers. In this limit, intrinsic states involving the genuine intrinsic normal modes are constructed and serve to estimate transition matrix elements. The method is demonstrated for the cases of spherical, axially-deformed, and triaxial overall shapes of the valence proton-neutron distributions.