A helpful soul contacted me on IRC today: it turns out that the google cache for my homepage shows linkspam.
There is no linkspam just viewing the page. There is also no linkspam if I wget the page using the googlebot useragent.
The common denominator seems to be that we all use DreamHost for web hosting. I submitted a ticket with DreamHost yesterday: their support staff responded that this must not be a DreamHost issue, and it must be a problem with Google. I am extremely disappointed with DreamHost right now.
Has anyone heard of similar linkspam that only shows up for the Google webcrawler, but not for normal visitors? What other ways could it detect the googlebot, other than via useragent?