  It is thought that magnetic fields must be present in the interiors of stars
to resolve certain discrepancies between theory and observation (e.g. angular
momentum transport), but such fields are difficult to detect and characterise.
Asteroseismology is a powerful technique for inferring the internal structures
of stars by measuring their oscillation frequencies, and succeeds particularly
with evolved stars, owing to their mixed modes, which are sensitive to the deep
interior. The goal of this work is to present a phenomenological study of the
combined effects of rotation and magnetism in evolved stars, where both are
assumed weak enough that first-order perturbation theory applies, and we focus
on the regime where Coriolis and Lorentz forces are comparable. Axisymmetric
"twisted-torus" field configurations are used, which are confined to the core
and allowed to be misaligned with respect to the rotation axis. Factors such as
the field radius, topology and obliquity are examined. We observe that fields
with finer-scale radial structure and/or smaller radial extent produce smaller
contributions to the frequency shift. The interplay of rotation and magnetism
is shown to be complex: we demonstrate that it is possible for nearly symmetric
multiplets of apparently low multiplicity to arise even under a substantial
field, which might falsely appear to rule out its presence. Our results suggest
that proper modelling of rotation and magnetism, in a simultaneous fashion, may
be required to draw robust conclusions about the existence/non-existence of a
core magnetic field in any given object.
