Ok, it's not shiny and polished, but then I don't really need to start a tool restoration hobby. I wanted a usable tool.
I took it apart, brused of the dirt with an old toothbrush, and soaked it in vinegar overnight.
A little more work with a toothbrush, some 320 grit carbide paper, and some machine oil, and voila:
I bought a new blade, as even after spending a half hour with 120 grit paper, I was making little impression on the back of the old blade. I could sharpen it enough to plane pine, but it was looking like a career to flatten the back properly so that I could get it scary sharp.
I priced a new one at about €85 + postage, so I guess this is a bargain.
The base is flat. Or at least, sufficiently flat that I cannot tell the difference.