While my faction change processed, I passed the time thinking about how much of a butthead Garrosh is. And what about the Alliance's arrival in Pandaria? I wanted to experience it. Well, Blizzard had me covered. As soon as I made my way to Stormwind, I received the quest to head out to Pandaria on the Skyfire. After zoning into Pandaria, I immediately fell to my death because the Skyfire disappeared from under me.
I assumed my quest had ended there because I'd already been introduced to Pandaria through the opposite faction. My slight OCD for quest completion pushed me to visit Wowhead, where I discovered that I could complete the Alliance-side starter quests after all. Woo!