  Here, we show that spacetime modulations provide an exciting route to realize
complex nonreciprocal couplings, and in particular the elusive axion response.
We develop an analytical formalism to homogenize anisotropic spacetime crystals
in the long wavelength limit. It is found that spacetime crystals with suitable
glide-rotation symmetry can have a giant axion-type response, several orders of
magnitude larger than in natural materials. The nonreciprocal axion response
may have interesting applications in optics, for example in electromagnetic
isolation, and in addition may enable exciting new forms of light-wave
interactions.
