Odyssey doesn't have a disk based cache at all, so you can't empty what's not there

It just caches the pages to memory.
History and cookies are other things and not related to the page cache itself, so your question is set a bit wrongly. Cookies can be deleted with Odyssey's cookie manager ("Windows->Cookies..." menu entry) or by deleting the database file (conf/cookies.db).
Other things Odyssey can save to different database files (if you have configured it to save them):
conf/History.db - Browsing history information
conf/Passwords.db - Saved passwords, can be managed from "Windows->Passwords..." too
conf/TopSites.db - Statistics from the pages you've visited, writes a lot there
conf/WebpageIcons.db - Small icons you see on tabs or quicklinks etc
There's also HTML5 related LocalStorage directory where pages can save certain amount of their own data.
You can freely delete any of these files to clean things up. I myself don't like to collect anything to these permanently when browsing, so I've disabled saving of cookies, history, passwords, topsites, and website icons. I have LocalStorage enabled for the compatibility, but I've made a softlink for its directory to point to ENV:, so it gets cleared at every reboot. When you keep things clean, browser doesn't slow down during the years by the amount of collected data, and no need to worry about privacy that much
