I remember from childhood that you say "Hare Rama" first.
If you know Hindi, you can see the following video where a Sankarachrya talks about it:
https://www.youtube.com/watch?v=yVe3SSfJk0g
When did Gaudiya Vaishnavas reverse the order of the Mahamantra? Do non-sectarian Hindus and non-gaudiyas chant in the traditional order?