  Photons carrying a well-defined orbital angular momentum have been proven to
modify spectroscopic selection rules in atomic matter. Excitation profiles of
electric quadrupole transitions have been measured with single trapped
$^{40}$Ca$^+$ ions for varying polarizations. We further develop the
photo-absorption formalism to study the case of arbitrary alignment of the
beam's optical axis with respect to the ion's quantization axis and mixed
multipolarity. Thus, predictions for M1-dominated $^{40}Ar^{13+}$, E3-driven
$^{171}Yb^+$ and $^{172}Yb^+$, and B-like $^{20}Ne^{5+}$ are presented. The
latter case displays novel effects, coming from the presence of a strong photon
-- magnetic dipole coupling.
