A number of 3-quasiparticle isomers have been found and characterised in the odd-mass, neutron-rich, 187Re, 189Re and 191Re nuclei, the latter being four neutrons beyond stability. The decay of the isomers populates states in the rotational bands built upon the 9/2-[514] Nilsson orbital. These bands exhibit a degree of signature splitting that increases with neutron number. This splitting taken together with measurements of the M1/E2 mixing ratios and with the changes observed in the energy of the gamma-vibrational band coupled to the 9/2-[514] state, suggests an increase in triaxiality, with γ values of 5°, 18° and 25° deduced in the framework of a particle-rotor model
