= Solution
Place the trigonal axis along $z$, the axial ligands on $\pm z$ and the equatorial ligands in the $xy$ plane. \b[The ordering is $e''(d_{xz},d_{yz})<e'(d_{xy},d_{x^2-y^2})<a_1'(d_{z^2})$.] The $e''$ pair has nodal behaviour along both the equatorial ligand directions and the axial axis, minimizing direct ligand repulsion. The in-plane $e'$ 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.
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