Today, the thermostat worked fine when I checked it in a pot of boiling water. That seemed odd. I went to put it back in the motor, and, when I did, the seat fell out. Usually they are stuck in pretty good and have to be pried out. I noticed, as I worked, [the seat] kept falling out--that's when it hit me. The seat is designed to wedge into its orifice and stay in place.
At high engine RPM, the water pressure pushes the spring loaded thermostat off the seat, the water bypasses the thermostat, and the motor runs cool. As the motor slows down the thermostat re-seats and warms up the powerhead up to operating temperature.
For whatever reason, this seat would not stay put. When the water pressure unseated the thermostat, the seat came loose and jammed the thermostat in an unseated position. I had a new thermostat kit handy, so I pirated another new seat and used it with the old thermostat. It works fine. Now I know to stock up on seats and change them before I waste money on an expensive new thermostat.