The low-lying states in the odd-odd and unstable isotopes (106,108)In have been Coulomb excited from the ground state and the first excited isomeric state at the REX-ISOLDE facility at CERN. With the additional data provided here the pi g(9/2)(-1) circle times nu d(5/2) and pi g(9/2)(-1) circle times nu g7/2 multiplets have been re-analyzed and are modified compared to previous results. The observed gamma-ray de-excitation patterns were interpreted within a shell model calculation based on a realistic effective interaction. The agreement between theory and experiment is satisfactory and the calculations reproduce the observed differences in the excitation pattern of the two isotopes. The calculations exclude a 6(+) ground state in (106)In. This is in agreement with the conclusions drawn using other techniques. Furthermore, based on the experimental results, it is also concluded that the ordering of the isomeric and ground state in (108)In is inverted compared to the shell model prediction. Limits on B(E2) values have been extracted where possible. A previously unknown low-lying state at 367 keV in (106)In is also reported.