That does look peculiar. Everything in the cache folder should be files that mod_pagespeed tried to rewrite. There are two ways that I know of that this can happen:
1) You reference some third-party resource (say an image from another domain, or google analytics script) and you have explicitly enabled rewriting of that domain with ModPagespeedDomain www.example.com or ModPagespeedDomain *.
2) If your server accepts HTTP requests with invalid Host headers. Try (for example) wget --header="Host: www.fbi.gov" www.yourdomain.com/foo/bar.html. If your server accepts requests like that it may be providing mod_pagespeed with an incorrect base domain, and then subresources would be fetched from the same domain (so if www.yourdomain.com/foo/bar.html references some.jpeg, and your server accepts invalid host headers, we could fetch www.fbi.gov/foo/some.jpeg as the resource). There was a recent security release that makes sure all of these subrequests are done against localhost (not arbitrary third-party websites). Please see: https://developers.google.com/speed/docs/mod_pagespeed/CVE-2012-4001
You might want to look through these folders and see what specific resources are in there. I think that the biggest concern you should have is that someone might be trying to perform an XSS attack on your users or maybe a DDoS attack against another website (like www.fbi.gov), using your server as one vector. I do not think that these folders are indicative that your server itself is compromised.
If you would like to discuss this more, https://groups.google.com/forum/?fromgroups#!forum/mod-pagespeed-discuss is a good list to join and email.
