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I use PuTTY on machines I regularly use and I'd thought of just changing the port it connects to but, as I said above, I was hoping not to have to sacrifice one service (httpss) so another (ssh) can use it's port.
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To scan packets to have a program proxy your request to another is suicide. Your router is not likely going to be able to do it (unless of course you have a sophisticated one), and at any rate, the service/routr/firewall will be too bogged down by trying to filter requests that it won't be worth it.
That's likely one reason SysAdmins at large corporations try to limit their firewall rules to the bare essentials.
WeirdX is an X-Windows JAVA client that can run in your browser. That may or may not suffice.
Are you sure you cannot run your web server under port 81, 8080, 8000 or something similar? Why not try accessing your web server through another proxy service such as The Cloak? They should be able to bring you the pages you request over any port to you no problem.