What, if anything, would happen if you:
took two 20" box fans
took off the front grills
taped them together face-to-face
sealed off any other openings except the back grills
and turned each one on at the same time at the same speed?
Hmmmmm…would it rise up and then tip over and break?
Cecil would say that it’s your duty as a Doper to actually make the experiment rather than just posing the question.
Of course in practice turning them on at exactly the same time and at the same speed would be difficult to guarantee, but damn the torpedoes.
Mr. Blue sky I’m with Chef Troy on this. Do the experiment then report back on the results. Well, at least have somebody on standby to tell us what hospital or funeral home we can go to visit you. 
What’s the matter Mr. Blue Sky? Bored today? Stuck at home with nothing to do?
Don’t cross the streams!
Actually, I had just finished replacing the CPU fan on computer at work and remembered seeing a car show in which the physics of how a torque converter works (my mind is weird that way).
While I don’t have two 20" box fans, I DO have two 14" round fans, but they don’t have the metal housing, only wire mesh ones.
Actually, making them both turn on at once wouldn’t be hard. Find a power outlet that has a switch. Turn the switch off, turn both fans on and plug them into the switched outlet(may need an adapter). Flip the switch and watch the fireworks!
Enjoy,
Steven
OK, I took the two 14" fans and removed the front grills and faced them together. Unfortunately, the blades stick out too far to clamp them together. There did not seem to be anything weird going on other than the spot of dead (that is, unmoving) air at the center of the hubs.
I did try altering the speeds. If I turned fan #1 to high and fan #2 to low, fan #2 began to slow down. I stopped the experiment before it went on too long, however, since I’m not ready to sacrifice either fan to science. I may seek out some second hand fans or, at least, some really cheap ones.