GND vs COM? Which one should I connect the ground to?
I would recommend finding the schematics for this. Because it could be the speaker outputs. Attaching to the tape head circuit without modifying it might cause failure due to the line signal coming out of the phone is several times greater than the signal that is produced on a tape head. Also there are other things, like cassette present switch, actuated by the basket assembly, that without it defeated, it wouldn't allow the unit to switch to the &quot;cassette&quot;.
Other things you should be aware of, is you need to use coupling caps between the phone and the stereo, mainly because phones don't isolate their signal, and a small amount of voltage is present on the signal lines on the phone jack.
COM is the signal common, or signal ground (which is not a correct term). It is different than DC ground or GND. These are different things. COM refers to a common to an AC signal (in electrical power, it is the same as the neutral or white wire) Some circuits tie DC GND to Signal COM, some don't have them referenced at all.