Jetta no reverse

There's really nothing in the linkage that would go that would not interfere with the forward gears. It sounds like you have fine fluid pressure in the transmission if the forward gears work ok, but the loss of reverse sounds like either a solenoid in the valve body is not working (which may set a fault in the transmission control module) or that something is blocking the fluid flow (some of the channels in the valve body are tiny and it would only take a grain of debris)
I would say that your first step would be to have the modules scanned. This is the least expensive and least intrusive procedure.
After that you may want to have them drop the transmission pan and inspect and see of there is excessive debris in the pan, indicating the failure of one of the clutch packs or other internal wear- It's usually OK to see fine "glitter" in the pan, but NOT OK to see "metal flake" or any chunks. If everything looks good, it may be worth it to try changing the fluid completely and seeing if that solved the problem- it takes more pressure to activate reverse than the other gears, so if your fluid is low it won't build enough pressure.
If none of the above work, then the problem is likely farther internal- it could be the valve body, but that's an expensive part to "test" since you can't replace it, and it's usually more feasible at that point to simply go all the way and replace the transmission.