I think the divide between left and right has become too great to reconcile: this country needs to be split into two parts, and believers for large government/taxes/regulations can move to one, believers for limited government/taxes/regulation can move to the other.
I would go so far as to propose even three countries.
1. The existing USA
2. A progressive USA
3. A tea-party USA
National vote on a state-by-state basis.
I would suggest that a constitutional convention be convened by the most ardent supporters of both the tea party and progressive movements. Each would write up their ideal constitution, without consideration of their opposing sides. The progressives could add in rights to healthcare, immigration, gun control, and so forth. Tea partiers could add in rights to property, limited taxation, individual rights and so on. Then each state would have a vote. If a state had at least 60% votes in favor of a new constitution, it would join the new union. Otherwise, it would stay in the continuing United States.
Doing this would allow states such as California and New York to join a “progressive” US, while states such as Texas, Wyoming, and South Carolina could join a “tea party” US. However, states with split viewpoints would continue in the existing US.
Following referendums would be held every two years for the next decade.
Within a short time, we would see how the new unions would prosper, attracting migration, and even additional states. Individual states could even split into parts
(North and South California) if they so chose do so.
The current system has become so bipolar that it is pointless to seek reconciliation.