There is no guarantee in a purely libertarian society that the economic and cultural values of the society will become either left or right wing. But that's the point. Libertarianism inherently isn’t right wing or left wing.

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Without an authoritarian government to enforce their beliefs, the left and right are effectively meaningless in both their cultural and economic contexts.

As government power decreases, culture and economics leave the domain of coercive establishment institutions and enter a state of spontaneous order where there is no left or right, only freedom.

Therefore, the political compass *probably* looks more like this. As you get more libertarian, it becomes more impossible to be left or right wing.

You can definitely identify yourself as a left-libertarian or a right-libertarian, but in its purest form, libertarianism is not a left or right wing ideology.