The field inside the cage would be flat, and there would be a gradient of zero (unless you were also generating fields with some power source inside). I think you could look at the surface of the cage as reflective, both for radiation that originates inside it and radiation that originates outside it. You could say any wave induces a current that induces another wave, so it reflects; or you could say it was a reflector, and there is another virtual universe on the far side of it, like you can think of mirrors in optics.
There is a bit of a head-scratcher for me, though - if a wave causes a current which reradiates another wave, is the current so close to the surface that the wave is only reradiated back out on the same side of the cage wall? If the current circulates throughout the depth of the cage wall, why aren’t there equal radiations coming out of both surfaces (inner and outer)? I guess the current must circulate at so shallow a depth that the radiation can only be out of the same surface. I feel ashamed, too, because I should know this, but maybe the coffee hasn’t worked yet…