We analyze atomic photoexcitation into the discrete states by twisted photons, or photons carrying extra orbital angular momentum along their direction of propagation. From the angular momentum and parity considerations, we are able to relate twisted-photon photoexcitation amplitudes to their plane-wave analogues, independently of the details of the atomic wave functions. We analyzed the photo-absorption cross sections of mixed-multipolarity $E2-M1$ transitions in ionized atoms and found fundamental differences coming from the photon topology. Our theoretical analysis demonstrates that it is possible to extract the relative transition rates of different multipolar contributions by measuring the photo-excitation rate as a function of the atoms position (or the impact parameter) with respect to the optical vortex center. The proposed technique for separation of multipoles can be implemented if the targets atom position is resolved with sub-wavelength accuracy, for example, with Paul traps. Numerical examples are presented for Boron-like highly-charged ions (HCI).
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