The "Unnatural" section here covers a few varieties of unnatural reverb effect [0]. (I'd also recommend many of the items with the "theory" tag on this blog)
In addition to those, reverbs that modulate the pitch signal are very common in music, but are physically impossible on a large scale. Most are subtle and create a detuned "thickening" sort of sound. Some will change the pitch by a whole octave or more. [1]
Artificial reverbs can also have their decay rates adjusted dynamically, and can even decay "negatively" so that the reverb actually grows in volume. Any other parameter could also be adjusted in real-time (depending on the reverb) [2]
Convolutional reverb means you convolve the impulse response of a system with a "dry" input signal to get your output. If the impulse response has any content before t=0, that content will cause a sound in your input signal to have an effect on the output before it happens. "Acausal" is the jargon.
that's possible with normal reverbs too. reverse reverb is quite a popular creative tool. i've used it on snares to make them suck you in before popping you out
