The general structures of the vibrational states of HCO and its isotopomer DCO are analyzed in terms of periodic orbits (POs) and continuation/bifurcation diagrams. Both bound and resonance states are considered. It is shown that the members of the pure overtones are guided by POs, even in the continuum. In particular, it is demonstrated that the highly anharmonic states localized along the dissociation coordinate correspond to orbits originating from saddle-node bifurcations in the classical phase space.