Hello, Jeremy.
I've sent an email to WordPress.com support about the result.
I've updated WordPress to 3.6 release version, reinstalled Jetpack and tried connecting to WordPress.com. As we expected, it succeeded in connecting without any error.
WordPress 3.5.2 -> failure
WordPress 3.6 beta -> success
WordPress 3.6 -> success
One thing is clear: there was a problem not in the server configuration or PHP settings, but in some other places we couldn't specify.
Error was gone at any rate, so I mark this topic as resolved.
Thank you for your help.