We present extensive pseudopotential density functional theory calculations dedicated to analyze the stability, electronic properties, and structural isomerism in Cu6 clusters. We consider structures of different symmetries and charge states. Our total energy calculations reveal a strong competition between two- and three-dimensional atomic arrays, the later being mostly energetically preferred for the anionic structures. The bond lengths and electronic spectra strongly depend on the local atomic environment, a result that is expected to strongly influence the catalytic activity of our clusters. Using the nudged elastic band method we analyze the interconversion processes between different Cu6 isomers. Complex atomic relaxations are obtained when we study the transition between different cluster structures; however relatively small energy barriers of approximately 0.3 eV accompany the atomic displacements. Interestingly, we obtain that by considering positively charged Cu6+ systems we reduce further the energy barriers opposing the interconversion process. The previous results could imply that, under a range of experimental conditions, it should be possible to observe different Cu6cluster structures in varying proportions.