Nuclear collective states are usually grouped into sequences of energy levels called bands. However, the definition of a such band is not unique and model dependent because of different inter-band transitions which does not allow to order the energy levels sequentially. On the other hand, collective models allow for approximate determination of physical nature of considered excitations, e.g. vibrational excitations, rotational motion and so on.

Nowadays, there is considered a possibility of existence of nuclei with
exotic,e.g. tetrahedral or octahedral, symmetries. This hypothesis opens up a new possibility of classification of nuclear energy bands. It turns out that, in principle, the energy levels can be grouped into bands of levels generated by sub-Hamiltonians having different symmetries. In this case the global Hamiltonian has the symmetry which is a common symmetry of all
sub-Hamiltonians.

This idea allows to explain a puzzle of coexistence of energy bands with
different symmetries. For example, it is expected that, in many nuclei, the g.s. rotational band has axial or D_{2h} symmetry (for gamma unstable nuclei), but other bands can have either higher, e.g. tetrahedral, or lower symmetry than the ground state band.

The gadolinium and dysprosium nuclei are considered as examples.