 Solution

Place the trigonal axis along $z$, the axial ligands on $\pm z$ and the equatorial ligands in the $xy$ plane. The ordering is $e^{\prime \prime }(d_{xz},d_{yz})<e^{\prime }(d_{xy},d_{x^2-y^2})<a_1^{\prime }(d_{z^2})$. The $e^{\prime \prime }$ pair has nodal behaviour along both the equatorial ligand directions and the axial axis, minimizing direct ligand repulsion. The in-plane $e^{\prime }$ pair points into the equatorial region. The $d_{z^2}$ lobes point at both axial ligands and its torus also samples the equatorial ligands, giving the largest repulsion in the regular point-charge geometry.

Rotation by 120° and reflection in the equatorial plane preserve the geometry; they enforce the indicated doublet degeneracies. The ordering concerns a regular trigonal bipyramid with equivalent ligand-field assumptions, not every distorted five-coordinate complex.