  In the mixed-mode asteroseismology of subgiants and red giants, the coupling
between the p- and g-mode cavities must be understood well in order to derive
localised estimates of interior rotation from measurements of mode multiplet
rotational splittings. There exist now two different descriptions of this
coupling: one based on an asymptotic quantisation condition, and the other
arising from coupling matrices associated with "acoustic molecular orbitals".
We examine the analytic properties of both, and derive closed-form expressions
for various quantities -- such as the period-stretching function $\tau$ --
which previously had to be solved for numerically. Using these, we reconcile
both formulations for the first time, deriving relations by which quantities in
each formulation may be translated to and interpreted within the other. This
yields an information criterion for whether a given configuration of mixed
modes meaningfully constrains the parameters of the asymptotic construction,
which is likely not satisfied by the majority of first-ascent red giant stars
in our observational sample. Since this construction has been already used to
make rotational measurements of such red giants, we examine -- through a
hare-and-hounds exercise -- whether, and how, such limitations affect existing
measurements. While averaged estimates of core rotation seem fairly robust,
template-matching using the asymptotic construction has difficulty reliably
assigning rotational splittings to individual multiplets, or estimating mixing
fractions $\zeta$ of the most p-dominated mixed modes, where such estimates are
most needed. We finally discuss implications for extending the two-zone model
of radial differential rotation, e.g. via rotational inversions, with these
methods.
