Yes I do.

OK. Not saying you haven't done this, but check the bulbs. Also make sure the sockets have a good ground. With the "jumper" in place on the trans connector, and the key on, see if you have 12 volts at the socket.
If you don't have 12 volts at the socket, you will have to back track the wireing. Could be at the bulkhead connector, or if I remember correctly, there is a connector in the driver "kick pannel". Should be a 8 wire connector, and the backup lite wire is white with a tracer.

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Charrlie_S
65 Valiant 100 2dr post 170 turbo
66 Valiant Signet 170 nitrous
64 Valiant Signet
64 Valiant 4dr 170
64 Valiant 4dr 225

So what would be my options if the mechanism inside the transmission is the one not working?

Take a ohm meter, and connect it across the two outside pins of the switch. Put the trans in reverse, if you show continuity (low ohms) the switch/trans is not the problem.
You said you had 12 volts at the connector, but no lights when you jumpered the outer wires on the connector. That leads me to believe you have a wireing problem between the switch and the BU lites.
If you want to give me a call, send me a PM, and I will send you my Cell number.

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Charrlie_S
65 Valiant 100 2dr post 170 turbo
66 Valiant Signet 170 nitrous
64 Valiant Signet
64 Valiant 4dr 170
64 Valiant 4dr 225

Thanks for your concern, Charlie.
Quick question. According to the schematic, one of the wires from the bulkhead goes to the connector for the radio. I have no radio. Could this possibly be the source of my problems?

Do you mean, there is no radio, or it doesn't work? The radio and the BU lites are fed from the same fuse, but if you have 12 volts, at the BU lite switch, that is not your problem. How ever you could have a bad connection at the bulkhead connector, where the power comes from the switch to the lites. It might be better if you could call me, from a cell phone while you are at the car.

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Charrlie_S
65 Valiant 100 2dr post 170 turbo
66 Valiant Signet 170 nitrous
64 Valiant Signet
64 Valiant 4dr 170
64 Valiant 4dr 225

There is no radio. The connector is just dangling there.

I have found the problem. I fixed the problem, only to find another one.
Charlie, thanks for all your help. Wouldn't have found that first problem without advice.

Well, what was it?

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Charrlie_S
65 Valiant 100 2dr post 170 turbo
66 Valiant Signet 170 nitrous
64 Valiant Signet
64 Valiant 4dr 170
64 Valiant 4dr 225

The connector in the kick panel wasn't making good contact with the connector coming from the fuse block. I took the little brass prongs out and sanded them down. Turned the key, put it in reverse, and nothing. so I tried the paper clip thing, and the lights came on. The problem is in the transmission then.

OK. New problem. Back up lights turn on when safety switch connector is connected with a paper clip, which led me to believe that the problem is in the transmission; i.e. the rooster comb. Crawled under, changed filter, band adjustment, rooster comb is fine. I can't figure it out.