Thanks, Kyle.
I think we have this one solved. Your help is very much appreciated!
Here's the system information from my Joomla site:
PHP Built On Linux p3nlhg475.shr.prod.phx3.secureserver.net 2.6.32-604.30.3.lve1.3.63.el6.nfsfixes.x86_64 #1 SMP Mon Oct 26 11:56:49 MST 2015 x86_64
Database Version 5.5.43-37.2-log
Database Collation latin1_swedish_ci
Database Connection Collation utf8mb4_general_ci
PHP Version 5.4.19
Web Server Apache
WebServer to PHP Interface cgi-fcgi
Joomla! Version Joomla! 3.5.1 Stable [ Unicorn ] 05-April-2016 22:45 GMT
Joomla! Platform Version Joomla Platform 13.1.0 Stable [ Curiosity ] 24-Apr-2013 00:00 GMT
User Agent Mozilla/5.0 (Windows NT 6.1; WOW64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/51.0.2704.103 Safari/537.36
Under GLOBAL CONFIGURATION and SEO Settings, "Search Engine Friendly URLs" was set to YES. Setting this to "NO" solved the problem.
As I mentioned previously, this is a Rocket Theme Template site using Gantry as a foundation. Nothing much else on it, though. So, whether the SEF is 3rd party or not, I don't know, but it's something to look into. For now, I'm happy to see the login and registration working.
Thanks again,
Sly