I turned them with a wrench for maybe 20 minutes, with no sign of tightening. So I ran the water while continuing to turn them. To my surprise, the water alternated between flowing and stopping as i turned the valves in a single direction. So these valves--which are different than the ones on the other sinks--seem somewhat like a binary switch, if that's possible.

It's called a ball valve. The other kind is a gate valve.

The handle would have stopped you from turning it more than a 1/4 turn. See the little nub above the turning part? It would have caught on something on the handle.

Here's a diagram of a ball valve (in case you're curious how it works). This one is a straight through design instead of the right-angle you have, but it's a similar idea.

When the hole in the ball is lined up with the pipe, water flows. When it's turned 90 degrees, the water stops.

Makes perfect sense. Thank you!

Yes you have a 1/4 turn valve. If you have missing handles. You can buy a little handy tool called a 4 Way Water Key. They can be gotten for as little as $5.00 and can come in handy when needed.

It's not clear from your picture whether the handles (knobs) have been removed or were broken off at some point. If you just occasionally need to turn the water off, a pliers should be fine to turn the valve 90 degrees. The part that's missing is likely specific to the particular brand and model, and may be hard to find. These valves are inexpensive ($5-10), so it may be easiest to just shut off the water supply, remove the old valves, and put on some new ones.

Valves are cheap; just replace them. SharkBite angle stops