It seems that something about the way I have jumped between Cycles and Blender render has caused issues with a file. I am new to render engines and not that experienced in Blender in general, so I have been disorganized about things and have ended up having to jump around.
Right now I want to do up this model with a diagram feel, so I switched from Cycles back to Blender render, in order to do up some flat materials and make them shadeless and such. I switched off nodes on the materials, and when there seemed to be issues, I deleted all the materials thinking maybe redoing them would help. The new materials have been assigned to the individual faces that have them.
The model renders pretty well. (It has a few issues due to quickie geometry but I can live with them for now.) But in Materials, Texture, or Solid view there are a lot of faces that turn black or xray - they make the things behind them in the same object invisible, as well as themselves. Recalculating normals has helped in some spots, but not others. In the case of the mirrors, selecting the mirrored faces and flipping those normals had no effect, or made the face x-ray. I checked for non-manifold geometry, and that isn't the cause. I get the feeling it is a normals issue, maybe I'm wrong.
I do have a version of this file from before the Cycles materials were added, but I wanted to know if there is a way to correct this.