I tried various things before finding this article on the Apple support site that explained how to change a tag a music video appropriately (see the bottom of the page). Basically, it's a setting in the Info -> Options pane, thus it's not really hidden, but I had trouble finding it. I had troubles because it's not an option that you can set for multiple files at once (which I was trying to do). So I thought it was worth sharing.
Another inexplicable limitation is that you can't designate videos as TV Shows. I'm really hoping Apple improves the behavior and options for sorting video content in iTunes 6.1 (or 7, with the way that they seem to be numbering them now).

