After replacing the hose valve in the garage, I promised to repair its leaky upstream shutoff valve:

I shut off the hard water supply, dismantled the valve, and let everything soak in a cup of white vinegar for a few hours. The fizzing was a wonder to behold and the parts came out much cleaner without any effort at all.
Removing the handle required the handle puller and considerable rapping on the corroded-in-place handle at the tapered shaft. That reddish disk used to be a tin-plated steel data plate, but now it’s just a corroded sheet:

Because a shutoff valve will be open nearly all the time, it has a large washer that seals the cap and valve stem in addition to the usual stem packing:

Attempting to remove the screw from the stem broke the head into two pieces:

Worse, the screw shaft was a soft mass of corroded brass, so I had to drill it out and chase the threads with a 10-32 tap. I replaced the full-open washer with a slightly smaller one from the supply box, which required drilling out the hole to suit, adding some packing string under the main cap, and replacing the packing around the stem. But, eventually, putting everything back together works fine with no leaks at all.
This turned out to be slightly less horrible than I expected, which probably doesn’t justify procrastinating until the evening before the coldest night of the season.