No, he did not.
He took credit for what the house did.
He himself did not move rightward.
He was forced by political reality to sign legislation he didn’t want to sign, and with a smile on his face. That is what you call moving rightward. Not as a matter of heart obviously but as a matter of action, which is what matters. Point being that politicians are influenced by their situation. Which undercuts your 1:1 extrapolation from how Romney governed in MA.