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We formulate criteria for identification of the nuclear tetrahedral andoctahedral symmetries and illustrate for the first time their possiblerealization in a Rare Earth nucleus 152Sm. We use realistic nuclear mean-fieldtheory calculations with the phenomenological macroscopic-microscopic method,the Gogny-Hartree-Fock-Bogoliubov approach and the general point-group theoryconsiderations to guide the experimental identification method as illustratedon published experimental data. Following group-theory the examined symmetriesimply the existence of exotic rotational bands on whose properties thespectroscopic identification criteria are based. These bands may containsimultaneously states of even and odd spins, of both parities and paritydoublets at well defined spins. In the exact-symmetry limit those bands involveno E2-transitions. We show that coexistence of tetrahedral and octahedraldeformations is essential when calculating the corresponding energy minima andsurrounding barriers and that it has a characteristic impact on the rotationalbands. The symmetries in question imply the existence of long-livedshape-isomers and, possibly, new waiting point nuclei -- impacting thenucleosynthesis processes in astrophysics -- and an existence of 16-folddegenerate particle-hole excitations. Specifically designed experiments whichaim at strengthening the identification arguments are briefly discussed.

Spectroscopic Criteria for Identification of Nuclear Tetrahedral and Octahedral Symmetries: Illustration on a Rare Earth Nucleus