I don't think left = govt is very accurate at all. The origins of the terms right and left come from the wings of the French parliament prior to the French revolution.
The right wing supported the monarchy, inequality, tradition and the left wing believed in enlightenment theories of equality, democracy, etc.
I think the right and left division, stemming from this post Enlightenment debate, is fundamentally a division between those who presuppose that all people are equal (left) and that all people are not equal (right).
Similarly you might have left libertarians who believe that all people are equal and should have a greater amount of liberty or right wing anarchists who believe that all people are unequal and believe that this will be reflected given liberty.
You can have Nazis who are right wing and do not believe in human equality but don't believe in liberty either. Or communists who believe in human equality but still share the same disbelief in liberty.
The main political dividing lines are (imo)
Right vs left (inequality vs equality as a desirable state of man)
Liberty vs totalitarianism
Centralization of power vs decentralization