What would cause a gas gauge to work backwards? When the tank is full it registers empty and when empty it registers full? All suggestions appreciated.

maybe you have the wires reversed on the back of the dash gage? Has it always been doing this backwards reading stuff?
Or did you change out some wires?
Those two little shunt coils on each side of the needle are un-balanced by the amount of resistance sensed by the float arm variable resistor and which ever one gets more current swings the needle that one way and visa-versa. This type of circuit allows your gas gage to work without passing any current into the tank unit.

Give us some background information. Has any changes been made to the system lately? When did the problem first show up? With this information we can give a more accurate answer.

Have the wires to the battery been reversed?

When I purchased the car ghe gauge did nothing. I replaced the dending unit in the tank. It has been backwards since.

Silly old me, I never asked what year model car!

Did you check how the tank unit worked before you put it in the tank. Was it a 30 ohm unit? I know it isn't easy to do. I use a 10' piece of wire from the dash unit to the tank unit and a ground wire to simulate the chassis ground.
When I replace one of the tank units I run an extra ground wire from one of the bolts to chassis ground to be sure it is grounded. I also replace the wire from the gage to the dash.
The Filling station has a note about the units for a 1930 unit.

Mr. Mack. It is on my 1954. No, I did not test it before I installed it. I purchased as a new one for a 54. I assumed, which I should never do, that it was OK. First chance I get I'll drop the tank and start over. Got some other things to do first. Thanks for your input.

It is possible that the sender unit is working in reverse (high ohms at the wrong position).
Tony