  Rotation is typically assumed to induce strictly symmetric rotational
splitting into the rotational multiplets of pure p- and g-modes. However, for
evolved stars exhibiting mixed modes, avoided crossings between different
multiplet components are known to yield asymmetric rotational splitting,
particularly for near-degenerate mixed-mode pairs, where notional pure p-modes
are fortuitiously in resonance with pure g-modes. These near-degeneracy effects
have been described in subgiants, but their consequences for the
characterisation of internal rotation in red giants has not previously been
investigated in detail, in part owing to theoretical intractability. We employ
new developments in the analytic theory of mixed-mode coupling to study these
near-resonance phenomena. In the vicinity of the most p-dominated mixed modes,
the near-degenerate intrinsic asymmetry from pure rotational splitting
increases dramatically over the course of stellar evolution, and depends
strongly on the mode mixing fraction $\zeta$. We also find that a linear
treatment of rotation remains viable for describing the underlying p- and
g-modes, even when it does not for the resulting mixed modes undergoing these
avoided crossings. We explore observational consequences for potential
measurements of asymmetric mixed-mode splitting, which has been proposed as a
magnetic-field diagnostic. Finally, we propose improved measurement techniques
for rotational characterisation, exploiting the linearity of rotational effects
on the underlying p/g modes, while still accounting for these mixed-mode
coupling effects.
