My .htaccess file is exactly the same (default, with no modifications to what WordPress installs) as it is on my other sites that work.
I will try again today with inmotion (after a long long wait late last night my chat with support just disappeared in mid-conversation), but in the meantime, I wanted to make note of a pattern I've noticed about this problem.
I can go to any page in the admin interface EXCEPT those that are flagged as needing updates, the updates page at update-core.php, the themes page at themes.php, and the plugins page at plugins.php. All of these pages are present in the admin folder, but any attempt to reach them results in a 404 error.
I don't understand the inner workings of WordPress that well, but this certainly looks like a clue to me. Could it maybe be something that really isn't the fault of my hosting at all?