The lowest positive- and negative-parity bands of and neutron-rich even-even Ne isotopes are investigated using a theoretical framework based on energy density functionals. Starting from a self-consistent relativistic Hartree-Bogoliubov calculation of axially symmetric and reflection-asymmetric deformation energy surfaces, the collective symmetry-conserving states are built using projection techniques and the generator coordinate method. Overall a good agreement with the experimental excitation energies and transition rates is obtained. In particular, the model provides an accurate description of the excitation spectra and transition probabilities in . The contribution of cluster configurations to the low-energy states is discussed, as well as the transitional character of the ground state. The analysis is extended to and the shape-coexisting isotope , and to the drip-line nuclei and . The role of valence neutrons in the formation of molecular-type bonds between clusters is discussed.