I am a computer controls engineer. While I know nothing about fuel level sending units, I felt it only proper to add my worthless two cents, since "I wasted a perfectly good hour listening to Car Talk".
An interesting question to ask would have been "does the gage go to full instead of empty when you turn the engine off". If it goes to full, then I would have stronger support lords the gage itself being bad.
I suspect that the fuel gage is no more than a voltmeter, and the fuel level sending unit uses a rotary potentiometer wired as a voltage divider. To do this, you place 12 volts on one side of the potentiometer, and ground (also known as 0 volts) on the other side. These 2 terminals connect to opposite sides of a resistive coating. There is then a 3rd terminal, called a wiper that taps into the resistive coating at different spots. When the wiper is at the ground end, we see 0 volts on it. When the wiper is on the 12-volt side, we see 12 volts on it. If the wiper is 1/2 way in between, we would see 6 volts on the wiper.
The catch is, if you were to swap the 0 volt and 12 volt connections on the potentiometer, it would read 12 volts on empty, and 0 volts on full. Therefore, I think that an improperly wired fuel level-sending unit caused the problem.
In reality, they probably don't feed the gage with 0 to 12 volts. They probably use a precision voltage reference, which reduces the supply voltage down to a constant 5 volts regardless of the system voltage. This would prevent the gage from moving due to system voltage variations.
There is probably a bunch of filtering taking place in the gage, so it responds to changes slowly. This would make the meter not respond every tome we go over a bump or stop sharply.
I could also go along with it being an improperly installed fuel level-sending unit, if there is a mechanical way to install it wrong so the potentiometer moves in the reverse direction.
I guess I got even, now that you wasted some perfectly good minutes reading my stupid comments. Seriously though, I enjoy your show on WFYI Indianapolis, and this story was of high entertainment value.