  We formulate criteria for identification of the nuclear tetrahedral and
octahedral symmetries and illustrate for the first time their possible
realization in a Rare Earth nucleus 152Sm. We use realistic nuclear mean-field
theory calculations with the phenomenological macroscopic-microscopic method,
the Gogny-Hartree-Fock-Bogoliubov approach and the general point-group theory
considerations to guide the experimental identification method as illustrated
on published experimental data. Following group-theory the examined symmetries
imply the existence of exotic rotational bands on whose properties the
spectroscopic identification criteria are based. These bands may contain
simultaneously states of even and odd spins, of both parities and parity
doublets at well defined spins. In the exact-symmetry limit those bands involve
no E2-transitions. We show that coexistence of tetrahedral and octahedral
deformations is essential when calculating the corresponding energy minima and
surrounding barriers and that it has a characteristic impact on the rotational
bands. The symmetries in question imply the existence of long-lived
shape-isomers and, possibly, new waiting point nuclei -- impacting the
nucleosynthesis processes in astrophysics -- and an existence of 16-fold
degenerate particle-hole excitations. Specifically designed experiments which
aim at strengthening the identification arguments are briefly discussed.
