Now that it's apart, I took the time to clean everything before I reassembled. To reassembled it, here's a couple of diagrams.
My awesome version, which, after the lower gasket thing comes the strut.
Once you put everything on the strut in the right order, you need to tighten the top bolt on before you can move on. Here's the trick. Pull the dust boot up from the bottom:
And grip the strut shaft with a pipe wrench.
FIY, this is a bad idea. If you marr the surface of that strut shaft, it will in time destroy the seal on the strut housing, blowing the shock. That's the reason for the dustboot; even particles of dirt can damage that seal. At the very least, put a rag between the pipe wrench's teeth and the shaft, or tape up the teeth. There's a hex socket at the top of the strut shaft:
The correct way to do this is to hold the shaft with a hex key while tightening the nylock nut. There's a specialized socket you can buy that allows the hex key to pass through, and it has a provision to tighten the socket with an adjustable wrench instead of a ratchet. Some people use an air ratchet instead; the ratchet tightens the nut faster than the shaft can spin.
Thank you Dudesky, however, my shafts are unmarred.