So I have a pair of ST-6600 integrated shifters for a double crankset, and I've encountered a problem I haven't seen before. The left shifter works perfectly when no cable is installed. It feels crisp, and you can see that where the cable head goes (not exactly sure what it's called) moves cleanly to both positions as it should. However, once a cable is installed, I can upshift once and the shifter will not downshift. The cable head is properly seated where it should be. There doesn't appear to be any obstruction in the shifter, and at least visibly everything looks fine. I have given the shifter a good WD-40 flushing and oiled it as best I could. This is with a brand new, Shimano shift cable, and it happens whether the cable is under tension or loose.

I am stumped. I have come across STI's that simply no longer work, but not ones that seem to work fine without a cable. Does anyone have any ideas?

The shifter must be in the small ring position before attaching the cable. Flip the smaller shift lever several times before attaching the cable with no slack. See parktool.com for shifter installation and adjustment instructions.
Failing that, try flooding the shifter(s) with WD40. This can be done without removing the shifter.

That's what I did. The problem is that once the shifter is upshifted to the big ring, it won't shift back down if the cable is installed. The cable is properly seated where it is supposed to be. I have also already flushed the shifter with WD40 and oiled it.

Without a cable, the gear where the cable head goes into is pulled by weak spring when the ratchet is released (downshifting). Inserting a cable introduces drag, given that a steel cable is not that flexible enough to bend on the small radius of the gear without resisting bending. To overcome this, the FD's stronger return spring is the one that pulls on the cable to return it to the original position.

Insert the cable and pull on the cable when you downshift. Then try installing the shifter and the FD and see how it goes. It's probably just a very weak spring on the shifter.