Couldn't get PHP mail() to work. My problem was this:
phpinfo() showed that the SMTP parameter was set to 'localhost' in php.ini, which I thought should have worked.
Changing this to the explicit IP of the SMTP got mail() working! I don't have direct access to php.ini, so at the top of my email function I added:
ini_set('SMTP', '10.xx.xx.xx'); // -where 10.xx.xx.xx represents the correct explicit value.