In the end, it appears to be some issue with Internet Explorer's caching. It seemed to retain some caches for the sites in Favorites. There were actually two solutions to this:
a) For the user, the fix was to delete the link from Favorites and then re-add the link.
b) In code, we added the following tag to our index.jsp
to force IE9 into IE9 document mode, as per this question:
<meta http-equiv="X-UA-Compatible" content="IE=9">
These solutions are independent and can be used one at a time, or both at once.