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We classify all electromagnetic interactions of axions and axion-like particles, showing that there are in general four different axion-photon couplings possible instead of the only one normally considered. We find that the novel couplings arise naturally in the models which contain heavy magnetic monopoles or dyons and explain why the latter models are well-motivated from the theoretical viewpoint. We show that the new couplings of axions lead to a significant modification of conventional axion electrodynamics. We derive the generalized axion Maxwell equations and discuss their phenomenology, in particular setting new targets for axion experiments. We find that a future detection of an axion or axion-like particle with certain parameters can serve as circumstantial evidence for the existence of magnetically charged matter.