Do progressives/regressives respond the same to cognitive dissonance?

No they do not. Chis Mooney discusses a study that answers the question definitively. Which of course is supported by other studies like this one. So cognitive dissonance is not the same for both political leanings, as the very apt terms progressive and regressive indicate. In this article Mooney shows that the more educated the regressive, the stronger the avoidance of cognitive dissonance! The opposite is the case for progressives.