So as not to damage the cabinet, I would remove the woofer and put my hand inside the cabinet and push the tweeter out.
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The tweeter is just stuck because its been in the same spot for years. Get a couple of really small allen/hex wrenches. Put the short ends of the allen wrenches in 2 screw holes opposite each other. Straighten the allen keys out so they are pointing towards you and the small ends are behind the face of the tweeter. Pull gently on both wrenches at the same time and it should pop right out.
If for some reason that doesn't work, crazy glue something to the face of the tweeter that you can use as a handle, and pull it out that way.
Hey everybody: I actually came across a more elegant (and cheaper!) solution. Fix the tweeter in place (accidentally) by loosening/tightening the three inner torn screws. The distortion went away! I called Madisound (who sold me the tweeters) and told them what I'd done, and they said it made sense in terms of the design of these tweeters. Seems I fixed it without having to replace anything!
Thanks, though, for all the thoughtful suggestions!