If your shift cable runs underneath your bottom bracket thru a little plastic guider piece, make sure it's screwed down tight. Also, get your back wheel spinning and take a look at the gears and see if they are wobbling. I had a broken back axle and it ended up bending my hub and now I need to replace the whole shebang because the wobble is so bad...

Anyway, like you said, my original question is still unanswered:  Why is the chain still skipping?

(I just flipped the chain into the fourth orientation and it still slips.  Like I said, I realize the chain and gears need to be replaced anyway, so I'm going to do that soon.)

I suppose one possibility is that the master link has been altered enough via disassembly and reassembly that it's what's causing the slippage.  I may try removing the master link and clipping the chain together with a chain tool.

At least that's what I read somewhere -- I haven't tried it.  I also haven't read it *everywhere* so it makes me wonder how true it is.  But it would explain what you're seeing ...

EDIT: Found it.  From Mountain Bike Peformance Handbook --

Chain Tip From Wayne Stetina, Shimano North, American R&D Manager :
If you remove the chain when it is only halfway worn out and flip it over, you will double your chain life.  In other words, your chain will be turned inside out.  The other side of the rollers will now contact the gears, and the derailleurs will now be laterally bending the chain in the opposite direction.  Stetina says that Shimano engineers discovered this phenomenon quite by accident.

... seems  to make sense to me, but again, haven't tried it.  But it doesn't explain how it's skipping in three of the four possible orientations (though I'm not sure there's more than two unique orientations that matter), as at least one of those three must be the original orientation.  Unless you made a mistake, and did the same thing twice ...

I bought a new chain to see if I could get away putting a new chain on while keeping the gears, but it's a no-go.  The chain slipped in my little test ride down the block and back.  So I put the old chain on again.  Now, the old chain slips like crazy.

How could that be?  My thought was that I put the chain on in a different orientation from before.  There are four possible orientations for a chain to be on a bike.  The left side of the chain can be flipped to the right side, and interior side of the chain loop can be flipped to the outside.  For a new chain, the orientation is irrelevant since one can assume that the chain is symmetrical in both ways.

Now I've tried flipping my chain inside to out, but the chain still slips.  I then flipped the chain so that the left side is now the right side (or vice versa) but the chain still slips.  I find it hard to believe that the two orientations work in combination, or that both orientations each alone significantly affect chain slipping, but I guess that I'll have to test that out next.

If I flip the chain this one last time, I'll have tried all four possible orientations.  If it slips after this, I'll be a bit dumbfounded.  Why would taking the chain off and putting it back on cause the chain to begin slipping?

* I didn't do any chain cleaning inbetween.
* The chain has a master link for removal/replacement.

In any case, I plan to replace the drive train in the next week or two.