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The possible existence of shape-coexisting nuclear configurations withtetrahedral symmetry is receiving an increasing attention due to unprecedentednuclear structure properties, in particular in terms of the exotic 4-foldnucleonic level degeneracies and the expected long lifetimes which may become anew decisive argument in the exotic nuclei research programs. The presentarticle addresses the rotational structure properties of thetetrahedrally-symmetric even-even core configurations coupled with a singlevalence nucleon. We focus on the properties of the associated Coriolis-couplingHamiltonian proposing the solutions based on the explicit construction of thebases of the irreducible representations of the tetrahedral point-group on theone-hand side and the microscopic angular-momentum and parity projectionnuclear mean-field approach on the other. It is shown that for one-particleoccupying an orbital belonging to the $E_{1/2}$ or $E_{5/2}$ irreduciblerepresentation, the rotational spectrum splits into two sequences, thestructures analogous to those of the $K=1/2$ rotational bands in the axiallysymmetric nuclei. Although the spectrum is generally more complicated forone-particle occupying a 4-fold degenerate orbital belonging to the $G_{3/2}$representation, an appearance of the correlated double-sequence structurespersists. The spectra of the doubly-magic tetrahedral core plus one-particlesystems can be well interpreted using the analytical solutions of the firstorder Coriolis-coupling Hamiltonian. We introduce the notion of the generalizeddecoupling parameters, which determine the size of the energy-splitting betweenthe double-sequence structures.

First order Coriolis-coupling for rotational spectrum of a tetrahedrally-deformed core plus one-particle system