[Exim] Incoming mail lost without trace

Hey:

I wonder if anyone here can help me with something ...

I have two domains hosted with my ISP, which according to them are
correctly set up to deliver mail via SMTP. However, mail addressed to
users at these domains neither arrives nor bounces.

For the first week it did bounce, and the headers showed that it never
got any further than my ISP's mail servers [where it was looping], but
for the past few days it's just been disappearing. There are no
indications in /var/log/exim/exim_*log that any of this mail has ever
reached my mail server.

I have the domains listed in local_domains, and mail to my standard user
account arrives via SMTP without any problems.

I've been trying to sort this out with my ISP for nearly a fortnight
now, and we've reached an impasse where they insist that everything's
set up correctly at their end, and I insist that mail is nevertheless
not arriving at my machine.

Does anyone have any idea what might be wrong at my end? Are there
situations in which Exim just drops mail without logging anything?

-[z].