  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
$|\mathcal{L}\rangle$, right-handed $|\mathcal{R}\rangle$, and planar
$|\mathcal{P}\rangle$ configurations. The presence and the impact of the planar
component is investigated as a function of the total spin for mass
A$\approx$130 nuclei with the valence h$_{11/2}$ proton particle, valence
h$_{11/2}$ neutron hole and the maximum difference between principle 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 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 of
the doublet, the contribution of the $|\mathcal{P}\rangle$ 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
($|\mathcal{P}\rangle$) and non-planar ($|\mathcal{L}\rangle$ and
$|\mathcal{R}\rangle$) subspaces of the full Hilbert space.
