Group content by taxonomy term, but then also list all terms in each row
I figured it out!

Instead of just adding Category as a single field in the View and trying to figure out how to configure that to show each Category for a Thing row but only one Category header per group, I have the Category field on there twice - one that does not display multiple in one row (for the headers), and one that does display them all within one row (to list out all Categories for a Thing).

I hope this possibly helps someone else!