I'm in a bind, with this valve taken out of the wall and the water supply turned off.
It looks like these pieces are all intact, nothing broken. The valve stem screws into what I would call the valve seat, but the valve seat doesn't have anything to keep it from spinning freely, so I am unable to see how the valve works reliably.
The stem is quite long, and its not easy to look inside the valve unit inside the wall. So I can't add anything to the description about that.
Underneath the upper end of the valve (the part that holds everything in place and is visible in the picture at the opposite end from the valve seat), there was a fiber washer. I was thinking it provided pressure on the valve seat to keep it from spinning, but the original washer wash not very elastic at all.
I was able put the valve back together with some guess work by putting some stem packing as shown in the photo below:
This didn't assure me that I had it properly put together, but it removed the urgency, as I now have a working valve.
In th mean time, I learned that this valve is probably the American Standard Version II Tub Shower Stem of the Re-Nu style. I found this on a restoration plumbing website. These are described as from Art Deco fixtures from the 1930s to 1950s, which puts it in the right period.
The restoration site supplies reproductions with a modification at the end of what I have been calling the "seat". They call it the "barrel". The modified barrel has a rubber o-ring at the end (on the outside).
I am not sure about stem packing, but repair kits show a part that may be for stem packing. I think this would be necessary, and I'm guessing the original stem packing disintegrated over many years.
If anyone can explain more fully and correctly how this valve should be repaired, I'll accept the answer.