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Three- and two-level mixing models are proposed to understand the doubling ofstates at the same spin and parity in triaxially-deformed atomic nuclei withodd numbers of protons and neutrons. The Particle-Rotor Model for such nucleiis solved using the newly proposed basis which couples angular momenta of twovalence nucleons and the rotating triaxial mean-field into left-handed$|\mathcal{L}\rangle$, right-handed $|\mathcal{R}\rangle$, and planar$|\mathcal{P}\rangle$ configurations. The presence and the impact of the planarcomponent is investigated as a function of the total spin for massA$\approx$130 nuclei with the valence h$_{11/2}$ proton particle, valenceh$_{11/2}$ neutron hole and the maximum difference between principle axesallowed by the quadrupole deformation of the mean field. It is concluded thatat each spin value the higher-energy member of a doublet of states is built onthe anti-symmetric combination of $|\mathcal{L}\rangle$ and$|\mathcal{R}\rangle$ and is free of the $|\mathcal{P}\rangle$ component,indicating that it is of pure chiral geometry. For the lower-energy member ofthe doublet, the contribution of the $|\mathcal{P}\rangle$ component to theeigenfunction first decreases and then increases as a function of the totalspin. This trend as well as the energy splitting between the doublet states areboth determined by the Hamiltonian matrix elements between the planar($|\mathcal{P}\rangle$) and non-planar ($|\mathcal{L}\rangle$ and$|\mathcal{R}\rangle$) subspaces of the full Hilbert space.

Three-level mixing model for nuclear chiral rotation: Role of planar component