Periapsis.org is hosted by DixieSys (which has great deals for web-hosting, by the way), but something they changed with their server configuration broke my installation of SquirrelMail, that I use when I’m not at home. The specific error came when attempting to send email to certain domains. Evidently, it’s an error message from qmail:
553 sorry, that domain isn’t in my list of allowed rcpthosts
Evidently, the root cause of the error is that by default, SquirrelMail doesn’t use SMTP Authentication. So turn it on in the config.php file, as detailed in this DixieHelp post. And that fixed all my problems.