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The obvious solution seemed to be "DirectoryIndex somefile.html" but when I uploaded a .htaccess file with nothing but that line I got a 500 error with either my.site.com/my-directory or my.site.com/my-directory/somefile.html.
Am I right in assuming that the server admin has restricted the directives I can use in my .htaccess? Is that common?
The solution turned out to be simple, I put a "RedirectPermanent index.html somefile.html" :)