  The possible existence of shape-coexisting nuclear configurations with
tetrahedral symmetry is receiving an increasing attention due to unprecedented
nuclear structure properties, in particular in terms of the exotic 4-fold
nucleonic level degeneracies and the expected long lifetimes which may become a
new decisive argument in the exotic nuclei research programs. The present
article addresses the rotational structure properties of the
tetrahedrally-symmetric even-even core configurations coupled with a single
valence nucleon. We focus on the properties of the associated Coriolis-coupling
Hamiltonian proposing the solutions based on the explicit construction of the
bases of the irreducible representations of the tetrahedral point-group on the
one-hand side and the microscopic angular-momentum and parity projection
nuclear mean-field approach on the other. It is shown that for one-particle
occupying an orbital belonging to the $E_{1/2}$ or $E_{5/2}$ irreducible
representation, the rotational spectrum splits into two sequences, the
structures analogous to those of the $K=1/2$ rotational bands in the axially
symmetric nuclei. Although the spectrum is generally more complicated for
one-particle occupying a 4-fold degenerate orbital belonging to the $G_{3/2}$
representation, an appearance of the correlated double-sequence structures
persists. The spectra of the doubly-magic tetrahedral core plus one-particle
systems can be well interpreted using the analytical solutions of the first
order Coriolis-coupling Hamiltonian. We introduce the notion of the generalized
decoupling parameters, which determine the size of the energy-splitting between
the double-sequence structures.
