The previous post described the fundamental working relationship between the two hemispheres. That dance is what makes us human. But it's not the only relationship, nor even the best known.
The most public working relationship is that of the left brain speaking and acting for the whole brain. As previously pointed out, speech is mediated by Broca's area in the left brain, and most physical interaction with the world (touching, grasping, manipulating) is performed by the right hand, which is controlled by the left brain. This is why McGilchrist termed the left brain the Emissary.
The role of emissary means that the left brain is in a privileged role. If you accept that the left and right brains work independently and process information in completely different modes, then we have to consider that the left and right hemispheres may respond differently to the same stimuli, and there may be a conflict between the two responses. That raises the question of which response will be acted on with body and speech.
This is part of a larger question. I have greatly oversimplified the operation of the brain up to this point in order to create an understandable narrative. In fact, there are many parallel operations taking place in the brain at any one time, in both hemispheres. The brain is continually responding to stimuli, and it is also continually "choosing" which responses to act on. The process of "choosing" is not conscious but a complex equation involving both valence and arousal, which are in turn affected by context. Valence and arousal are independent dimensions of neural activity. Valence can be thought of as emotional content, and arousal can be thought of as emotional impact. The "choosing" is a learned response to evaluating valence, arousal, and context in real time.
The right hemisphere "chooses" one (or a few) response(s) among the many possibilities to present to the left brain. (I suspect one or a very small number because the corpus callosum that communicates the choice(s) is relatively small as brain pathways go.) Then the left brain "chooses" among its many possibilities, which include the right brain response(s), what will be acted on by body and speech. This puts the left brain in its privileged role.
This privileged role is what has led us to think of the left hemisphere as dominant. When the left hemisphere is severely damaged, we lose much of our ability to connect with the world, so we seem from the outside to not be the same person or not even "there". When the right hemisphere is severely damaged, we may talk and behave normally, and the loss of right-hemisphere function may not be so obvious to those who don't know us well.
It is a running theme on the TV series House that "people lie", and this doesn't mean the doctor believes people always do it intentionally. Furthermore, most psychological research must also deal with this issue. Reeves and Nass describe in The Media Equation how psychological experiments for the last 60 years have had to deal with the unreliability of verbal reports. The issue is that the left brain is "choosing" what to say. Generally, the most honest (and useful) response to most questions will come from the right brain since it is grounded in reality. But the left brain may choose to answer based on one of the many filters and processes it contains, ignoring the right brain response.
So the words and actions the left-brain initiates may represent only a faction among the brain's thought processes. Depending on ones life experiences, the distance between what is said and done may be miles from what is really going on in the brain. This distance can range from normal diplomatic speech to full repression of feelings resulting from serious abuse. These are learned responses to a complex world. Typically that learning is seen as necessary for survival (or advancement) of the whole by the left brain. But there are times when those lessons are not in our best interests.
Underlying the comments in this post is the assumption that the left and right hemispheres can carry multiple disagreeing thoughts and feelings. In a healthy person, the disagreement is typically minimal or well handled by the "choosing" mechanism. When the multiple thoughts and feelings diverge too much and are not handled well by our "choosing" mechanism, people feel considerable distress. Very often the underlying thoughts and feelings can't be changed. Then our only hope is to learn to "choose" better, and that isn't easy.
The left and right brains don't know the same things. The right brain has memory of its grounded reality. The left brain has memory of the its constructed reality. There is some evidence that the right brain has access to what the left brain knows, but not vice versa. This is consistent with the way that the right brain has access to the entire visual field while the left brain only has access to part of the right visual field, and the way that the right brain is aware of the entire body and the left brain is aware of only the right side. If true, this access makes the right brain a more complete picture of who we are than the left brain, and this is why McGilchrist termed the right brain the Master.
The asymmetry of the left-right divide in communication makes thinking and talking about the divide difficult. As I mentioned in my second post, the left brain can feel threatened by the exposure of its role and methods. I will repeat that I am not trying to denigrate the value or methods of the left brain: for every inappropriate coping mechanism is embodies there was undoubtedly an existential crisis making it necessary.
Do not take anything I say to mean the two brains are competing. They are both doing their best to protect and nurture our self as a whole, but they have very different views of the world and very different structures with which to deal with that world.