Edit: Everything below is pretty much superseded by a new post.
php-fpm: /var/lib/php/session was owned by root:apache, so php-fpm couldn’t create session files. Fix was chgrp nginx /var/lib/php/session. For some reason, after doing yum update, /var/lib/php/session was changed back to root:apache, but that was only noticed on one system so far, haven’t observed it on others, but also haven’t looked for it.
nginx: By default, index.php files aren’t though of as index files. Fix is to add ‘index.php’ to the ‘index index.html’ line.