The possibility of shape coexistence within the selenium isotopes and the general features of their structures have been examined within the framework of the DDM, IBM-1 and IBM-2. New data on ⁷⁶Se and ⁷⁸Se were obtained from gamma -ray and IC electron transitions. Levels and transitions in ⁷⁶Se were studied following the beta decay of oriented ⁷⁶As which had been dissolved in iron and cooled to 10.4 mK. The measured anisotropies of gamma transitions provided the multipole mixing ratios of 18 transitions, and previously proposed substantial L=0 mixing in three first-forbidden beta transitions was confirmed. Gamma-ray directional correlations and the IC electron spectrum of transitions in ⁷⁸Se were measured following thermal neutron capture by ⁷⁷Se. The correlation experiments gave the spins of 18 levels and the multipole mixing ratios of the more intense gamma -ray transitions. The good resolution electron spectrum allowed E0 transitions to be identified and also allowed the evaluation of IC coefficients and of X(E0:E2) ratios for Delta I=0 transitions. The 2⁺ levels at 1.778 MeV (⁷⁶Se) and 1.996 MeV (⁷⁸Se) appear to have the features characteristic of the lowest mixed-symmetry state in a vibrational-like nucleus. The level energies, B(E2) ratios, multipole mixing ratios and X values are compared with the results obtained from the DDM and IBM. The comparisons between theoretical and experimental data are extended to even-even selenium isotopes in the range A=72 to 82.