Home page request returns 404

This is the same issue documented at https://drupal.org/node/1491748 but (as indicated in the comments there) it affects 7.x as well as 6.x. I'm starting this as a new 7.x issue in case 6.x is not being actively maintained. This is a real, ongoing problem for a number of production sites running 7.x (we're on 7.22, someone else I think was on 7.23).

Basically, the home page randomly gets cached as a Page Not Found and we don't know why. Clearing all caches fixes the issue, but in the meantime visitors see a 404 instead of the home page. This bug doesn't seem to affect other pages, and it doesn't appear tied to amount of traffic.

We're running on IIS7 with a MySQL backend, no server cache programs, not running Boost or any other contrib cache modules. Content caching is turned on, as well as caching for some blocks and views. This has been going on for months now, and we're stumped.