I set up some rewrite rules, and they didn't work at all. Instead of redirecting to the new URL, I just got a 404 error. IIS didn't seem to even notice that the rules existed.
As per http://learn.iis.net/page.aspx/467/using-failed-request-tracing-to-trace-rewrite-rules, I tried to turn on Failed Request Tracing to get more details, but found no option for Rewrite in the WWW Server Provider Property Areas. This made me wonder if something was not installed properly.
However, I found some indications online that this could be due to installing Url Rewrite Module after IIS. Therefore, I went and manually modified the C:\Windows\System32\inetsrv\config\applicationHost.config to contain the needed area. Still all I got was a 404 error, and in the trace, lots of details about the 404 error.
I thought, "Hmm, well why don't I just reinstall the Rewrite Module 2.0 from the ground up?" I loaded the Web Platform Installer and noticed that it was installed, but no way to remove it. I went to Control Panel > Programs, clicked "Turn Windows features on or off", and nothing related to Rewrite Module seemed to be there. Finally I thought to look under the standard programs list in Programs and Features. There it was: IIS URL Rewrite Module 2. I right click and choose "Repair". Immediately things started working as they were supposed to.
Talk about waiting until the last step to try the simplest solution. Ah well, such is life some days.