Hi,Welcome to Just Answer, My name isXXXXX will be helping you with your appliance issue today.
You need to replace the drive gear assembly,in order to do this you will need to remove compactor from under cabinet tip it down on it's side and remove the bottom cover, this is located right under the bottom cover, the main gear the big one in the picture is what usually cracks and breaks, we do a lot of these on different compactors including yours, you can order this at the blue link below.
http://www.repairclinic.com/PartDetail/Gear/269615?modelNumber=GCG1580P0SS in it comes with instructions, thanks and well wishes, Bryan
Will you say you hear the motor running but it doesn't come back up if the motor is running it definitely have something to do with the gears or possibly the Rams, the reverse switch wouldn't cause this problem because the motor still running in it it was still trying to go down the motor would overheat and trip your overload, these are the RAM rods, you need to take the top off and check the power not on the ramrod, it may have broken and is not going up the Rod, it's definitely mechanical and electrical, there is a link below to all the diagrams.
http://www.searspartsdirect.com/partsdirect/part-model/GE-Parts/Compactor-Parts/Model-GCG1580P0SS/0432/0110000?pathTaken=&prst=0&shdMod=GCG1580P0SS hope this helps thanks, Bryan
Your first thought may be correct...the motor turns and the large & small plastic gears turn but not the chain that drive the threaded ram rods. Does this sound like a cracked main gear at where the chain sprocket attaches?