  Using first-principles density functional theory calculations, we investigate
the hydrostatic pressure-induced reorientation of the Mn--F Jahn-Teller bond
axis in the fluoride cryolite Na$_3$MnF$_6$. We find a first-order isosymmetric
transition occurs between crystallographically equivalent monoclinic structures
at approximately 2.15 GPa, consistent with earlier experimental studies.
Analogous calculations for isostructural $3d^0$ Na$_3$ScF$_6$ show no evidence
of a transition up to 6.82 GPa. Mode crystallography analyses of the
pressure-dependent structures in the vicinity of the transition reveals a clear
evolution of the Jahn-Teller bond distortions in cooperation with an
asymmetrical stretching of the equatorial fluorine atoms in the MnF$_6$
octahedral units. We identify a change in orbital occupancy of the $e_g$
manifold in the $3d^4$ Jahn-Teller active Mn(III) to be responsible for the
transition, which stabilizes one monoclinic $P2_1/n$ variant over the other.
