A model of shape coexistence, based on an intruder pi-(h9/2)2 excitation, previously applied to explain low-frequency upbending phenomena in 171Re and 172Os, is extended to heavier Os and Re isotopes. A phenomenological three-band mixing analysis of the yrast bands of even Os nuclei and the 5/2+[402] and 9/2-[514] bands of the neighbouring Re isotopes is used to deduce the excitation energies of the proposed intruder configuration as a function of neutron number. These are shown to be qualitatively consistent with an (h9/2)2 excitation. The implications of this model for other nuclei are discussed.