It turns out that ImageMagick is really quite good at reading, writing, re-arranging, and otherwise mucking with PDFs. Unfortunately, you need to know the proper incantation, which can take much trial and error to figure out. So, for my own future reference:
Split A PDF Into Parts
$ convert -quality 100 -density 300x300 multipage.pdf single%d.jpg
The quality parameter is the quality of the written JPEGs, and the density is the DPI (in this case, 300 DPI in both X and Y).
Join JPEG Parts Into A PDF
$ convert -adjoin file*.jpg doc.pdf
Rotate a PDF
$ convert -rotate 270 -density 300x300 -compress lzw in.pdf out.pdf
This assumes a TIFF-backed PDF. The density parameter is important because otherwise ImageMagick down-samples the image (for some reason). Adding in the compression option helps keep the overall size of the PDF smaller, with no loss in quality.
Now, if I can just figure out how to make future me remember to look here…