  Starting with the Gaudin-like Bethe ansatz equations associated with the
quasi-exactly solved (QES) exceptional points of the asymmetric quantum Rabi
model (AQRM) a spectral equivalence is established with QES hyperbolic
Schr\"odinger potentials on the line. This leads to particular QES
P\"oschl-Teller potentials. The complete spectral equivalence is then
established between the AQRM and generalised P\"oschl-Teller potentials. This
result extends a previous mapping between the symmetric quantum Rabi model and
a QES P\"oschl-Teller potential. The complete spectral equivalence between the
two systems suggests that the physics of the generalised P\"oschl-Teller
potentials may also be explored in experimental realisations of the quantum
Rabi model.
