Here's a concept:

The code references this developer resource page: http://archive.is/Jd2IA

Page talks about creating a Ruby file/script that pings Twitter to clear the cache of so-called "Twitter Cards." This essentially pings Twitter, which then clears the cache for the Card Image in order to load the new Card Image.

Conclusion/Question:

Is there is a Ruby file on the Org's server that has their Twitter login information and is (maybe now was) accessible to the public (with a bit of know-how)?