I created a layout in Visual Studio and it appears with the rest of the themes in Sitefinity. However, on that layout page, I render two partial views and the associated controllers, even though they have been created in the appropriate folder with the appropriate names, are never called. Am I missing something? If this cannot be done, then my evaluation of this CMS is at an end, as that is a deal-breaker.
It turns out I had to perform a RenderAction for the view in question instead of the RenderPartial I was doing previously. It isn't very intuitive and I don't know why that one view required the RenderAction while the other views on that layout are fine with RenderPartial. I'm marking this post as "answered".