  We present a theory of magnetoelectric magnon-photon coupling in cavities
hosting noncentrosymmetric magnets. Analogously to nonreciprocal phenomena in
multiferroics, the magnetoelectric coupling is time-reversal and inversion
asymmetric. This asymmetry establishes a means for exceptional tunability of
magnon-photon coupling, which can be switched on and off by reversing the
magnetization direction. Taking the multiferroic skyrmion-host Cu$_2$OSeO$_3$
with ultralow magnetic damping as an example, we reveal the electrical activity
of skyrmion eigenmodes and propose it for magnon-photon splitting of
``magnetically dark'' elliptic modes. Furthermore, we predict a cavity-induced
magnon-magnon coupling between magnetoelectrically active skyrmion excitations.
We discuss applications in quantum information processing by proposing
protocols for all-electrical magnon-mediated photon quantum gates, and a
photon-mediated SPLIT operation of magnons. Our study highlights
magnetoelectric cavity magnonics as a novel platform for realizing
quantum-hybrid systems and the coherent transduction between photons and
magnons in topological magnetic textures.
