Stay tuned...the problem, finally discovered by Grant R. (who deserves a serious lollipop for Incident #1039390) is that I'm still using port 8399 for production deployment. I left it as the default instead of changing it to the recommended port 80 or 443.
Apparently, there is internal ESRI use of 8399, so my traffic couldn't be seen by the tech support people (and only them), while people all over the place outside my network could see it fine. That was baffling and frustrating.
Whew! I'll get back to you when it is deployed on the correct port.