I suspect there is a misconception that system 1 (intuitive) is mapped to the right brain and system 2 (deliberate) is mapped to the left brain.
The left brain is livewired to be competent in sequential thought while the right brain is livewired to be competent in parallel thought. One is egocentric and the other is allocentric. One is symbolic and the other is empathic.
One is reductionist and the other is holistic. One is noun-centric and the other is verb-centric. One emphasizes individuality and the other the collective. We can make many analogies about the dichotomy between the two hemispheres of the brain.
However, the analogy that system 1 resides in the left brain and system 2 on the right is incorrect. Both systems are system 1. System 2 is just a reflective instantiation of system 1. The brain always employs parallel mechanisms for its processing.
Any notion of sequential circuitry is an illusion. It is a similar illusion that we describe as consciousness.
We however do not lose our consciousness when our left brain is damaged. Indeed we are incapacitated in our ability to perform sequential action or understand language, but we do remain conscious.
Consciousness like system 2 implies the capability of being reflective of one's thoughts. The kind of reflectiveness in the right hemisphere is of the allocentric kind. The kind that is in the third person.
One can argue that Robert Kegan's model for adult development cannot happen without the reflectiveness of the right hemisphere. Adult development is a consequence of the egocentric-self expanding itself towards the allocentric-self.
The two hemispheres are livewired for different concerns. The fluidity of human thought is a consequence of the the interplay of these two. We would not be able to solve Bongard problems without the interaction of these two hemispheres.
A Bongard problem is a rule finding game. The objective is to find the one consistent rule that is self-consistent for the graphics on the left and the right. Furthermore, the left rule is the anti-thesis of the right rule. 
The left hemisphere will be very good at identifying patterns. Unfortunately, it will be incapable of find alternative patterns. It requires the right hemisphere to perform a search for alternatives.
Our system 2 reflective and deliberative process kicks in when we are validating our solutions.
But analogous to AlphaZero that makes system 1 judgments about policy and value, the interplay between the left and right hemispheres makes judgements about patterns and alternative patterns.
That is conceptual wholes and conceptual distinctions. That is symmetry and symmetry breakings.
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