First of all, the phrase “cognitive dissonance” is often misused. It is not the same thing as motivated reasoning. Cognitive dissonance happens when people are trying to hold different beliefs that don’t fit together well (for example, “the Twins are all set to be a contender next year” with “the Twins will lose 100+ games this year.”) Or when we notice that our attitudes and behavior are notably inconsistent (when we don‘t notice, or affect not to notice, that‘s called “hypocrisy”). It’s not a pleasant feeling, and people try to get rid of it.
Motivated reasoning is basically all the mental gymnastics people do to justify believing what they want, based on dogmatism, emotionalism and/or (often ego-driven) cognitive biases, when said beliefs have little or no apparent grounding in fact and/or reason. People use it to, among other things, quell cognitive dissonance. In some ways, a well-known phrase that could be a synonym for m. reasoning is “wishful thinking,” only you don’t accept that it’s wishful when it’s you that’s doing it.
(It’s possible that learned specialists would take issue with my definition, perhaps as too rough-and-ready. I certainly wouldn’t claim that I’m right and they’re wrong. I’m just trying to provide an indeed rough-and-ready description for practical understanding in socio-political contexts.)
Suppose I feel the need to internally justify a vote for Donald Trump for president. I fall back on unthinking, irrational, completely unsupported dogma that right-wing conservatism is good doctrine that produces good results. Emotionalism in that I will be gleeful to see people of color, assertive women, liberals, etc., get theirs. And I can call on all kinds of cognitive strategies like denial, rationalizing, confirmation bias, authoritarianism, groupthink, and so forth to convince myself that he is in reality knowledgeable, fair, honorable, kind, generous, and all that good stuff. Again, despite apparently infinite evidence to the contrary. All of these processes generally are, in real life, entirely or at least mostly subconscious.
By no means is motivated reasoning confined to the political right wing. On the contrary, it is as pervasive as the very air. We all do it. Including me. Trying to be aware that one is doing it is the first step to replacing it with the discipline of logic – that is, to really reason from fact.