Template incompatibility with 3.5 causes blank admin pages

I have a client with a custom WordPress template that I did not design, and after I upgraded WP to 3.5, the admin pages became blank after you log in. The admin menu is still there on the left, but the top bar and the body of the pages are blank. The original developer is MIA, so I’m stuck trying to fix this. I’ve gone through all the standard procedures of disabling plugins and such, and I’ve determined that the admin pages are only blank when the custom template is activated. I can only change templates by editing the database in phpMyAdmin. If one of the default templates is activated, the admin works fine, but then the site looks all wrong.

I’ve checked the PHP error logs and no error shows up. I’ve turned on Firebug in Firefox and that doesn’t tell me anything useful either. The only thing I’ve got to go on is that only two files in the template refer to the admin: classes.php, where most of the custom admin pages for the template seem to be defined, and comments.php, which has a “Logged in as” line. It seems pretty obvious that the problem is somewhere in classes.php, but there’s a lot in there and I’m not enough of a code guru to know what to look for. Can anybody suggest some possible common incompatibilities with the 3.5 upgrade that I might be able to search for, or some other way to narrow this down further?