What causes it to shift into reverse but not move?

First thing tomorrow, take it back to the shop that rebuilt it!!

This is caused by the gear oil,there is none in the gear box, so get a transmission fluid and fill the gear box with it, run the engine idle for some minutes and then try select the gear.

Have you checked the transmission/gearbox fluid is at the correct level ? This will also cause this. Hope this is of help.

no net neccessarly. try shifting the trans manually instead

on a hyundai you have a shifting sensor attached to the transmission that if it starts to go bad it will cause many problems, hard shifting, loss of power or no shifting.

The late shifting could be a control problem but the no reverse is not. These trans had a habit of stripping out the reverse sun shell, the metal from it may have gotten into the valve body causing the late shifting, but it really does not matter. If you have forward movement, but no reverse it has to come out. Sorry dude. You need a trans.

Reverse clutches are burnt. Clutch material is getting into valve body causing late shifts since valves can't move. transmisison will have to be overhauled. wish i had better news. sorry

What engine? Rear wheel drive ?Sounds like shift linkage or cable out of adjustment slightly. Have a look at where cable or rod attaches to transmission for adjustment. If not sure of what your looking for get back to me with info.

look for transmission shift interlock solenoid fuse.or a bad brake switch will cause transmission not to come out of park.

your linkage on the transmission must be off. try having someone move the shifter while you watch the linkage arms under the car on the trans to make sure its all intact. it could be a loose or broken bolt on the trans causing it to not disengage from reverse this will cause it to not start.