After watching AGT's Winchester 97 Armorer's Course DVD, I found out a bunch of cool stuff that I didn't know previously about this old warhorse. I also took the receiver down to individual parts and put it back together again, removing all 100 years of accumulated carbon fouling in the process (there's dirty, and there's caked).
There are a couple of buttons on the receiver. 1, above, deactivates the carrier lock, so you can cycle the action without lowering the hammer first. 2 is attached to the cartridge stop. There's an identical assembly on the other side of the gun. What the button allows you to do is unload the magazine tube without cycling the gun. Press one and then the other to eject a single shell. Depressing both buttons simultaneously causes the magazine tube to enthusiastically disgorge its contents. The front of the trigger guard does not appreciate such tender ministrations (and don't ask me how I know this). One at a time is definitely the way to go.
One unique* feature of this gun is that all the lockwork is built into the shell carrier. The entire assembly hinges on a large pin (3 in the above picture). Everything, including the hammer, swings down when the gun cycles.
On the bottom of the carrier is the sear, which interfaces with the trigger guard at the back of the reciever. It turns out that the trigger stop (somewhat visible above) functions as a timing screw as well. Because the gun lacks a disconnector, it will fire when the gun closes. If the timing is off, the hammer may fall when the gun is not fully locked up, which would be bad. Therefore, the angle of the trigger vs. the sear lever can be adjusted with this screw.
* I assume, since this is the only shotgun I've really handled, much less detail stripped.