All attempts to send email to myself resulted in replies which contained the error:
all relevant MX records point to non-existent hostsThanks to this entry on the PkgExim4UserFAQ, I was able to get a clue
A probable cause for this might be that all MX records for the offending domain point to site local or link local IP addresses, which are ignored by the dnslookup router to protect from misconfigured external domains. The default configuration has relaxed checking for domains that the local system is configured to allow relaying to, so adding the offending domain to dc_relay_domains will most probably help. Please note that this entry might be necessary anyway to bypass relay control for the domains in question.Well, I did the requisite test, to find this among the output
Please note that no domain on the public Internet should have MX records pointing to site local or link local IP addresses, so you might check your externally visible MX records.
If this doesn't help, try analyzing the output of exim -d -bt firstname.lastname@example.org
ignored host mail.xxxx.com [10.0.0.x]
Ignored host?? Clearly not the same as one that is non-existent. So... the first error was a lie.
All the relevant records pointed correctly to a very much alive and well host, but exim chose to ignore it because it was local.
In order to get around this, you have to follow the suggestion of Marc Haber and tell it that it is going to relay email for your local domain (which sounds like a very bad idea) in order to get it to work.
I don't know why they did it this way, but I'm posting this here to help others figure it out.