A couple of methods are possible (see the videos), but the explanation here is using the 'Advanced Flyaway' method. Swing out and make a half turn. Swing back to the platform. As you swing over the platform, sweep hard backwards (like a straight jump) so you body stays in front of the platform. As you swing back down, bring your legs in front of you in a set, then drive backwards and do a double front somersault to catch the stick. Just as for cutaways, Double Flyaway Halfs are also possible.
Thanks to Alexis letting us know about this trick.