The left subscribes to the French Revolution, whose guiding principles were "Liberty, Equality, Fraternity." The right subscribes to the American formula, "Life, Liberty and the Pursuit of Happiness." The French/European notion of equality is not mentioned. The right rejects the French Revolution and does not hold Western Europe as a model. The left does. That alone makes right and left irreconcilable.
The left envisions an egalitarian society. The right does not. The left values equality above other values because it yearns for an America in which all people have similar amounts of material possessions. This is what propels the left to advocate laws that would force employers to pay women the same wages they pay men not only for the same job but for "comparable" jobs (as if that is objectively ascertainable). The right values equality in opportunity and strongly believes that all people are created equal, but the right values liberty, a man-woman based family and other values above equality.
The left wants a world -- and therefore an America -- devoid of nuclear weapons. The right wants America to have the best nuclear weapons. The right trusts American might more than universal disarmament.
The left wants to redefine marriage to include same-sex couples for the first time in history. The right wants gays to have equal rights, but to keep marriage defined as man-woman. This, too, constitutes an irreconcilable divide.
For these and other reasons, calls for a unity among Americans that transcends left and right are either naive or disingenuous. America will be united only when one of them prevails over the other. The left knows this. Most on the right do not.