Three- and two-level mixing models are proposed to understand the doubling of states at the same spin and parity in triaxially deformed atomic nuclei with odd numbers of protons and neutrons. The particle-rotor model for such nuclei is solved using the newly proposed basis which couples angular momenta of two valence nucleons and the rotating triaxial mean field into left-handed |L), right-handed |R), and planar |P) configurations. The presence and impact of the planar component is investigated as a function of the total spin for mass A≈130 nuclei with the valence h11/2 proton particle, valence h11/2 neutron hole, and the maximum difference between principal axes allowed by the quadrupole deformation of the mean field. It is concluded that at each spin value the higher energy member of a doublet of states is built on the antisymmetric combination of |L) and |R) and is free of the |P) component, indicating that it is of pure chiral geometry. For the lower energy member of the doublet, the contribution of the |P) component to the eigenfunction first decreases and then increases as a function of the total spin. This trend as well as the energy splitting between the doublet states are both determined by the Hamiltonian matrix elements between the planar (|P)) and nonplanar (|L) and |R)) subspaces of the full Hilbert space.
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