Personally, the symbol does represent the gyms. What its doing on the building is simple, we finally get a proper headquarters in the games. The sign over the building was merely an advertisement that its around the corner, in the middle green open area of Hiun City. Its purpose is simple. You go there and register. Its also where you get the games watch. The starter you seemingly get as a present, most likely from your mom. I'm to believe this present is given to you for passing, and getting the chance to register for Isshu's league, as you travelled from one of 4 previous regions, and as they are unified but not with Isshu's you must register in a much more organized league. I'm also inclined to believe the league's headquarters also indicate where you will battle the E4, even if we have seen it early, as no one would expect it. And it would be nice to finally have a properly organized league, and motivation. As a side-note, I would like if the organization of the league means that you will be able to track your trainer battles, and beating certain trainers gains you the advantage of recognition.