Just after coming up with the idea, I walked into the Techshop wood shop and found a pile of thin plywood scraps in the communal scrap bin. I considered this a sign, and decided to get started right away. The rack is made up of pieces with interlocking slots (half-lap joints) cut on the table saw. There is a backer board behind the spice bottles to keep them from slipping out the back.
I'm quite happy with the concept - the shelf fits perfectly, and the spices are much more organized and accessible than before. Of course, there is one obvious problem - not all the spices have labels on the tops. As a temporary solution, I'm going to put sticker labels on the tops that need them. Then I'm going to make some personalized, re-usable spice jars with labels printed on the tops.