I know some who can, and they were given the choice to keep their original House and original wand, or to re-take both quizzes.
Here's something to think about.
The first question you are asked for is your name. If you put your name, that's what you'll be seeing. If you put your original name, such as noxquill11, you'll be seeing THAT on your profile.
I redid mine. I got Ravenclaw again, but Holly wood, Phoenix core, and slightly springy
Holly. Phoenix. Seriously????????