My car will barely move in reverse.

Okay, I think I know the answer to this but I have to ask: when was the last time the transmission fluid changed? If it has been some time, then it is possible that some debris has broken loose from an impeller turbine, for example, and has floated downstream to where it is blocking a key gallery or channel and it is keeping reverse from engaging.

The way this happens is that the debris floats along until it gets to the valve body. The valve body is a device that is full of galleries and check valves at their ends. When you choose a gear, the gearshift solenoids fire in a specific order to determine the correct gearing. Everything seems to be find in this area as your A6 is trying to reverse, however, it is essentially failing. By essentially failing, it is going into gear but not moving very much.

That indicates to me that everything in the transmission is working correctly to that point. However, since there is a constriction in the fluid flow through the valve body to reverse, the gear is not receiving the correct flow of transmission fluid and it is, as a result, working poorly or not at all.

This is a fairly straightforward fix. Of course, the transmission has to be dropped from the housing and disassembled so you can access the valve body which sits on the outside of the transmission but is not easily accessible without a partial disassemble. Once the valve body — a device with lots of channels and galleries that direct the fluid to the proper gear — is exposed you will probably see one or more channels/galleries with gunk in them, usually at the check valve so that instead of the valve opening and closing in response to demand, the check valve remains partially closed most of the time, only allowing a little fluid through at a time, causing your problem (a check valve consists of a lightweight ball-bearing and a light spring that closes and opens in response to the demands of the transmission’s gearing).

Your technician will run the affected galleries with a device that is like a pipe cleaner and will clear out the gunk and other debris and will also replace the checkvalve and spring, repeating this process until all the dirty channels are cleaned (a gear may have more than one gallery devoted to it). Once they are all cleaned and the valving is working again, the technician will button things up and refill the transmission with the proper fluid to the proper levels.

That should do it.