For some minor purposes your rating has changed and RHP is keeping track. If you were a subscriber and tried to enter a tournament, you'd actually see your 'p1200' rating slotted in at some other position in the list of names.
Also there's some way of searching for players that will let you stumbel across the 'real' rating, I forget what it is.
But the main thing is that your opponent's score after each game will continue to be calculated as if you are at p1200. Once you've finished 5 games the 'real' ratings you had will be revealed in the graph in your profile.