Ok, I am trying the old reinstall since I had only published one thing. Let me try and get the terminology right.
I published a PAGE (like the link you provided) in the Category “Things” on my blog. When I click on the “Things” Category on my Sidebar, I get a 404, from my web host, not from WordPress. I hope that clears it up, like I said, I am new to this. I didnt change any permalink structure that I know of.
