I had never actually uploaded them in the first place — I had used the control panel to do it for me using fantastico (some suite of mysql things).
That said, I gave it a go and downloaded the package, deleted the server versions of the folders you mentioned and replaced them with my freshly unzipped versions. Sadly, there was no change. I still can’t add categories and still can’t delete comments via the comments section of the wp-admin (it looks like it is deleting them, but when I reload it I see that they are still really there).
Any other ideas?