This was written by cPanel support to one of our correspondants, which echoes what we say in our per-IP diagnostic page:
Thank you for your patience while I've looked into this for you. Presently it looks like you have the option "Send mail from account's dedicated IP address" enabled in your Exim configuration. This causes mail for the server to reference it's own domain name and any other names in the /etc/mailhelo file in the HELO header when mail is sent. This doesn't provide you much if any benefit, because you only have a handful of accounts using dedicated IP addresses while you have several hundred using the shared IP address, so this will cause problems like what CBL reported to you when accounts on the shared IP addresses try and send mail. I would recommend disabling this option, as this will allow the HELO header to be sent just with the server's hostname. The server's hostname is generally the only thing you want in HELO headers, as any other entries can look like attempts to obfuscate the source of the mail which is generally a hallmark of spam.
You can disable this option in WHM > Service Configuration > Exim Configuration Manager > Basic Editor > Domains and IPs. You shouldn't see any other issues with the HELO file is you set this.
Generally it's a bad idea to use the "Send mail from account's dedicated IP address" option unless you have a dedicated IP address for each customer, and you've verified that each customer's email does in fact use the dedicated IP address for sending email. Since it provides so little benefit, we recommend that you don't bother.
In the past, many cPanel issues were caused by cPanel bug 59875, which was fixed around 2011/12.
In 2015 or so, the issue re-surfaced (as CPANEL-3984), and we're informed by cPanel (the source of the information below) that the fix is currently (2016/02/11) going through Quality Assurance before general release. This will appear in a future update of version 54. If you have a version previous to this, you should follow the workaround below.
NEW NOTE:
According to cPanel support (April 2016), a similar bug has surfaced (CPANEL-5340) that will only affect resellers in certain configurations that use Exim. cPanel is researching how to resolve this issue, and in the meantime, please use the (same) workaround below.
Disable "Send mail from account's dedicated IP address" and use the "Reference /etc/mailhelo and /etc/mailips" options as explained here.. The 2 files /etc/mailhelo and /etc/mailips may need to be manually modified in the mean time.