Just spent too much time on finding out why sessions didn’t work on a particular site. To be precise: the PHP session cookie was not sent.
Turned out that there was a line in the site’s .htaccess that also set a temporary cookie:
<FilesMatch "\.php$">
Header set Set-Cookie "myCookie=1; path=/;"
</FilesMatch>
All was fine when I changed set into add:
<FilesMatch "\.php$">
Header add Set-Cookie "myCookie=1; path=/;"
</FilesMatch>
Cheers.