>Let's say I wanted to allow bidirectional communication between an internet client and a host computer behind a router without having to configure port-forwarding on the router. What are the options to accomplish something like this?

So I found this very cool tool that does exactly this:

https://pagekite.net/

The fast, reliable way to make localhost part of the Web.
PageKite makes local websites or SSH servers publicly accessible in mere seconds, and works with any computer and any Internet connection.
It's also 100% Open Source.