I found it. A permission called "Manage Advanced Grading Methods" (managegradingforms).
I had a previous question about gradebook calculations, and a good Samaritan on this board responded with some great advice. However, the method advised had a side effect with the gradebook AND reviewing attempts. Since my exams are now a part of an "advanced grading method" and the trainers didn't have permission for advanced grading methods, they couldn't see anything in the gradebook I'd set up (because the only thing in there were calculations) and the review attempt was limited.
So my apologies -- what I thought was a general question was actually quite specific to my case.
I'm sure this won't be a common occurance for others, but at the risk of being redudant I'm going to copy this explanation into the post related to exam caculations, just in case it can help someone else.