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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 $ζ$. 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.