Hi Kovid,
Please, help :-)
Kindly download books
from here.
I have included two books. English original and processed through my hyphenation plugin. Lithuanian original, hyphenated and also original tweaked via included "tweak ebook" menu. "Originals" both are converted from EPUB and show ToC and chapters fine. Processed and tweaked do not.
Device is Kindle 4 non-touch with latest firmware.
Tried with version 0.9.22 of Calibre.
I know, that ToC and chapters work with AZW3, it is my primary used format. But have never tried tweaking before. I have also tried to explode original and modified books, OEB contains "toc.ncx" and no other obvious issues.