Transit

As the hint and graphic suggest, the first step in solving this puzzle was to untangle the map, treating loops as flexible string. Doing that gave something like this:

This looks a lot like the Olympics logo, which makes the right orientation clear. Each word, it turns out, maps to one Olympic sport; the hard ones all turn up with a good google search. Using the official Olympic names, the sports are as follows:

The first letter of each sport, when read in ring order with the major-stop dot before the minor-stop dot, spells “TARDIGRADE.”