OK, the entire point of this blog post is to save some other poor sod from the pain that I've just been through.
If you have a Windows Mobile device and get synchronization error 0x8503001c, your life is close to over. Trust me.
You can delete and re-create the sync partnerships between your PC and your Windows Mobile device as many times as you want, with whatever permutations of synchronization settings you choose, but it's not likely to help. Try it anyway, but don't waste too much time on it.
What you're going to do, at some point after you've realized that it's all hopeless, is reset everything to factory defaults and start over.
The first tool you'll want is Dotfred's PIM Backup tool, available at http://www.dotfred.net/. Get it and back your device up (NOT to your device memory!).
Reset your device to factory defaults. On my Dopod you hold both of the multi-function buttons (the "-" buttons) while poking the reset button with your stylus. Press the Send button when it prompts you to confirm. Yours is probably different.
Re-create the sync relationship between your phone and your PC.
Restore your backup.