The thermostat didn't work anymore, owing to a leaky valve. Fortunately, my local autoparts store actually sold a rebuild kit for this valve! It wasn't easy, and I ended up having to glue the mechanism back together, but at least it works now.

Hours of cleaning with a wire brush and a bit of black paint restored the heater selector mechanism (just what do you call this thing, anyway?) I also cleaned up and lubricated the cables.