I've played around with the video settings some more, and found some odd behavior.
While in a UFO recovery mission, I went to the video options "Other" tab and selected the "Medium" configuration. Exiting the options menu, all the textures were being rendered correctly. I went in and disabled a few options, like bilinear filtering, dynamic lighting, glsl shaders, etc. My framerate vastly improved and the textures remained in working order. It even lasted back to the geoscape, where I could view the items in all their textured glory in the buy/sell and soldier equip screens.
I exited the program and restarted it, and the textures were back to being completely screwy again. So, I once again opened up the options menu, picked the "Medium" preset, and this time disabled all the options I had disabled before while I was there. Once again, the textures were rendered correctly. I saved my game, exited, and restarted the program once more.
Again, the textures were screwy. I tried using the "Medium" preset (textures were restored!) and immediately exiting the game without making any other changes. Upon restart, the "Medium" preset was still in effect, but the textures were gone again.
I still have no idea what's going on, but I actually got some useful-looking lines in my console. Attached is the console log after loading the game, using the "Medium" preset, and deactivating every checkbox on the "Advanced" tab.