The Cullens(Edward, Alice, and Emmett) were Cullens but were not related to each other in the official story. But the Hales(Rosalie and Jasper) were in the official story because they were both blonde and looked a lot alike. Before Rosalie was changed, she was actually Rosalie Hale, but before Jasper was changed, he was Jasper Whitlock. He took Rosalie's last name to keep up the twin story, and also because Rosalie doesn't want this life and wants to keep as much of herself from before she was changed as she can because she liked her perfect life before she was changed into a vampire and can never die or have children.
Hope this helps!!!