I just solved the same problem. In my case, the replacement switch was poorly designed.
There are three wires going to the backup light/neutral safety switch. You can use a test light or volt meter to confirm you have voltage being supplied from the fuse to the switch by measuring the voltage at the connection corresponding to the white wire (ignition ON). While the connector is disconnected, you can check the rest of the system (assuming the white wire is hot)by jumping the outside connections on the connector (NOT THE CENTER!) with 12 ga. wire while an assitant (or a mirror) tells you your backup lights are on (be careful not to short the wire to ground, it will at least blow the fuse, cause sparks and other nastiness). If so, the switch is the culprit, If not, your troubles are in the wiring between the connector and the lights in the trunk.
If you pull the switch out, you'll see that the switch has a metallic pin in the very center (inside the transmission), that must be all the way OUT for the backup lights to work. Remove the switch with a 1" box wrench (you can get there from underneath, without jacking up the car even), plug the connector back on, and with the ignition ON, push the pin in and out. When the pin is out the backup lights should light up. If not, and jumping (see above) the connector worked, your switch is "no good".
If the lights light up with the pin out, but it doesn't work when installed on the transmission, it could be, as mine was, that the new switch is TOO LONG, and doesn't ever allow the center pin to come all the way out and light up the lights. I used a for-real Chrysler switch from the junkyard, and now all is well. Wish me luck returning the switch I bought.
If you are curious, you can see how it work internally by placing a mirror strategically and shining a flashlight off the mirror into the hole where the switch goes. You'll see a spot, when the tranny is in reverse, where the plastic has a dip in it to allow the pin to come all the way out. If the switch is too long, the pin never goes all the way out despite the dip, and the lights remain dark. You can also see how the neutral safety works too!
Here's an alternate solution if the switch is too long. Put a shim of some kind on the switch like, a washer, to push the switch out farther.
Check the backup light switch.
If the bulbs are the cause, remove the tail light assemblies and replace the bulbs; if they still don't work, replace the neutral safety switch, located on the front of the transmission.
Try checking the neutral safety switch. This switch is the one in automatic transmissions that makes the backup lights "flip" on when going from park to drive as you go through the gears. When my neutral safety switch shorted my backup lights stayed on constantly.
Check bulbs and switch
check fuse and bulbs...switch
There is a backup switch in the gearbox. When you shift into reverse, the gear box switch contacts the switch and lights the white reverse lights. If you find that the reverse lights don't work it could be that the switch is not being triggered by the mechanism inside the gearbox. This is providing that everything else is working; wiring,bulbs,etc.
well it seems to be a common problem it might just be the switch on the transmission
Check bulbs Check fuse Check switch
Fuse, bulbs, switch, wiring.
Check bulbs and switch
Bad bulbs, bad switch located either on steering column or on transmission
There is no adjustment for the backup light switch. The switch must be broken or the fuse is blown or the bulbs are blown or the wiring is damaged.