Thanks Chris, that's it exactly, and your post helped me find this page:
http://www.radioeng.co.uk/audio.html
It doesn't work as I expected, though, due to my (complete) misunderstanding of the m/s signal. I thought the mid channel encoded only sounds that are in the middle of the stereo field, completely ignoring sounds at the extremes, and the side channel did the opposite... Something like the picture I attached, if that makes sense. (i.e. the amount of the signal encoded in the mid channel is biased towards things in the middle and tapers off to nothing at the edges. Vice-versa for the side channel.)
But if the mid channel is just a sum of L+R, then obviously it carries ALL the audio, regardless of where it is in the stereo field.
I guess that leaves me with another question: Is it possible to do what I've described? Like, what if I wanted to get rid of all the hard-panned sounds from a mix? Is it possible to filter a stereo signal to ignore all the sounds that are in one speaker, but not the other?