@shawnhcorey Thanks a lot for the advice. It's not paranoia (well, maybe there's some of that, too). It's just that they're library files, they don't even have a she-bang, but some of them are executable and others, apparently equal, are not. They caught my attention because Perl files are given such strange permissions on install, and that led me into this rabbit hole of investigating how and why.
What I ended up doing was chmodding 644 all .pm and .pod files and letting ExtUtils do its thing (i.e., turning 644 into 444) during the install process. It seems to have worked fine so far.