15:37:15.971: >> 554-Bad DNS PTR resource record.<cr><lf>

The arrows pointing to the right denote the remote server's response/query (arrows pointing to the left are your server's responses/queries).

As Rolf says you need to ask your ISP to set a PTR (PoinTeR) record (a reverse lookup record which the opposite of an A Record).

As you are probably aware a DNS query for a name requests the A record so that the name (e.g. google.com) can be resolved to an IP address. A reverse record, known as a PTR record allows you to resolve an IP address to a name. Mail servers use it as a way to ensure the name the server announces itself as matches the IP address the mail message originated from.  

I need some help with an odd problem.  Messages sent to a gmx.com address via MercuryC are getting delivered but ones sent via MercuryE are not.  I would appreciate some help understanding what is going on.  I confirmed this by sending a message from Mercury (Ctrl+N) by each method. 

The entries in the MercuryE connection history window for the failure are:

Connection error during handshake with 212.227.15.10.

Connection error during handshake with 212.227.17.4.

12:19:51: processing job MO0000A6
Resolved MX for 'gmx.com' to 212.227.15.10
Connecting to 212.227.15.10
Connection error.
Attempting connection to alternate, 212.227.17.4
Connection error on alternate.

The associated MercuryE log entries are:

T 20181220 121951 5c1b6ad9 Begin processing job MO0000A6 from postmaster@mydomain.com
W 20181220 121952 5c1b6ad9 Error connecting to primary server '212.227.15.10'.
W 20181220 121953 5c1b6ad9 Error connecting to alternate server '212.227.17.4'.
E 20181220 121953 5c1b6ad9 Processing of job MO0000A6 incomplete or failed.
T 20181220 121953 5c1b6ad9 Job MO0000A6 processing complete.

Edit:  A subsequent test via MercuryE with session logging enabled logged this:

15:37:14.941: --- 20 Dec 2018, 15:37:14.941 ---
15:37:14.957: Connect to '212.227.17.4', timeout 400 seconds.
15:37:15.971: >> 554-gmx.net (mxgmx117) Nemesis ESMTP Service not available<cr><lf>
15:37:15.971: >> 554-No SMTP service<cr><lf>
15:37:15.971: >> 554-Bad DNS PTR resource record.<cr><lf>
15:37:15.971: >> 554 For explanation visit http://postmaster.gmx.com/en/error-messages?ip=70.62.88.210&c=rdns<cr><lf>
15:37:15.971: << EHLO mydomain.com<cr><lf>
15:37:15.987: >>
15:37:15.987: --- Connection closed normally at 20 Dec 2018, 15:37:15.987. ---

I've looked at the explanation link buts it's beyond my level of expertise.  We have a dedicated IP address but it's from Spectrum Business so could very well be in a block of address recorded as dynamically allocated.   I see the EHLO as mydomain.com and wonder if that is considered a fully qualified domain name.  That's fixable if it needs to be; I just don't know.

Thanks for any help.

Well, the error message says "http://postmaster.gmx.com/en/error-messages?ip=70.62.88.210&c=rdns". You can check the PTR record (rdns) for your IP address for instance here: https://mxtoolbox.com/ReverseLookup.aspx

A mail server is according to RFC required to have a reverse DNS record, but some sites have their own requirements. Many reject rdns names that suggest that the address is dynamically assigned, and others refuse all generic PTR addresses (such as 10.10.168.192.in-addr.arpa).

If you have a fixed IP address you should be able to ask your Internet provider to set a different PTR record for that address.