I got the speaker out and learned about the grille. The speaker is held onto an oval frame by four small nuts and bolts, one at end corner of the 6X9 speaker, and the grille is bolted from the other side to the frame by four larger nuts, which screw onto long studs at each corner of the grill. It's a bit awkward getting the two top grille nuts undone because you have to reach way up inside to the top of the dash where there's not much clearance. If you get the four large nuts undone the grille pops out with only a little coaxing. The grille, by the way, will never flex: it's solid steel or pot metal and weighs over two pounds.
I feel more confident about reinstalling the whole thing once the radio is rebuilt, now that I see how it all fits together, although getting the holes to line up when doing the lower dash may be tricky.
I'll see if I can get a picture to accompany this. It shows the back of the grille, the speaker with its oddly-shaped gasket and the radio (with its back off - I was curious).