  We show that, contrary to assertions in the literature, the main contribution
to the axion-photon coupling need not be quantized in the units proportional to
$e^2$. In particular, we discuss a loophole in the argument for this
quantization and then provide explicit counterexamples. Hence, we construct a
generic axion-photon effective Lagrangian and find that the axion-photon
coupling may be dominated by previously unknown Wilson coefficients. We show
that this result implies a significant modification of conventional axion
electrodynamics and sets new targets for axion experiments. At the core of our
theoretical analysis lies a critical reexamination of the interactions between
axions and magnetic monopoles. We show that, contrary to claims in the
literature, magnetic monopoles need not give mass to axions. Moreover, we find
that a future detection of an axion or axion-like particle with certain
parameters can serve as evidence for the existence of magnetically charged
matter.
