Using the latest version of W3 Total Cache 0.8.5.1 I found that turning Page Cache on stopped my special menu and page system working.
When users were not logged in they saw a certain home page and menu, but when they are logged in they are supposed to see a different home page and menu.
However, I found that when I visited the website while logged in, the page cache saved that version of the page for later, and when I visited in a different browser not logged in, the logged in version appeared!!!
Also, I was trying to use 'Wordpress Mobile Edition', a plugin that detects if you're using a phone and serves up a different theme if so. However, page cache interfered with this one as well. W3 Total Cache does offer a redirection mechanism to another domain but I'd prefer to have the different theme run on the same domain...