I'm going through our site analytics, and have a load of t.co URLs which were referrers to a promotion we were doing. I'm trying to figure out if there is a way to reverse those back to the original tweet where they originated, through the Twitter API or other means. I can't seem to find a good means to do this though, is there one?
This is not possible with the public APIs that twitter provides.
If I understand correctly you want to find a tweet that originally had a particular t.co link embedded. i.e. The t.co when followed resolves to your site, not the twitter tweet.