Transmission slips and then goes into gear in reverse only

Transmission is done. Nothing electronic will cause that, only mechanical

limited slip differentials are so designed that when one axle looses traction the pressure is applied through the side gears to lock up diff so that both wheels have to turn. If it slips in reverse then in will slip going forward . I would be inclined to look for a reverse gear problem in the gear box/ transmission. If it is an auto check for reverse solenoid operation.

If your fluid level is right you may have a bad transmission. you need to talk to a mechanic.

How many miles on this transmission?

Perhaps it is in severe pain, because you failed to identify it it by make and model!

Servo or band worn out, o.ring gone hard or clutch pack burnt out.

Definitely not good. The transmission is slipping internally due to worn gears and clutch packs. It will have to come out for rebuilding, IMO. Have a transmission shop check it out.
Good luck.

Transmission range sensor may be failing. I would test it for opens and shorts.

Transmission model is 4L60E,and the symptoms you are describing is a common failure on these transmissions all the way back to it's predecesor,the 700 R 4 model.The reverse input shell,splines into the low/reverse planetary sun gear.The reverse input shell has a collar with splines made into it,and these splines are prone to stripping,and sometimes the collar itself will break free from the shell.Without the shell functioning,no reverse,and this is likely what you are hearing.Aftermarket companies sell heavy duty,and standard duty,replacements.Eventually,you will lose the forward gears also.Good luck.

http://www.2carpros.com/how_to/transmission_service_4.htm
Popular Problems Checks
1. Car will not go into gear:

• Gear selector cable has failed
• Brake lock solenoid has failed not allowing the gear selector to move out of "Park"
• Excessively low transmission fluid (note: if car is operated for an extended amount of time with low fluid level the transmission will fail prematurely)
• Shorted electrical component not allowing the PCM to control the transmission. Example: shorted fuse
• Flex plate (flywheel) is broken completely not transferring engine power to the transmission.

2. Car goes into gear but then fades out of gear or is slipping while driving:

• Transmission fluid is low
• Transmission clutch discs or bands are worn out or burned
• Faulty transmission shift solenoid.

3. Car goes into gear but does not shift out of first:

• Blown fuse to the PCM controller
• Faulty vehicle speed sensor (VSS)
• Shorted second gear control solenoid
• Faulty transmission controller (PCM)

Tips About Common Problems and Fixes