Hrm, very odd: I made no updates, changes, or reconfigurations of my server’s Apache daemon. The default httpd.conf has the rewrite module unloaded, but for some reason that was working fine under WordPress 4.3. Upgraded to 4.4, and no more rewrite without that module enabled.
As soon as I enabled the module in the site’s httpd.conf, the Permalinks started working again.
Interesting.