If we subdivide the line between left and right long enough, there will be a hundred factions and splinters on each side, so I will only consider the main, more visceral groups for this. I think most people fall into a moderate category.<quoted text> I hope you do not mean to imply that there is an equivalence. The phrase sounds catchy but what do you really mean? It all depends on how one defines right and left of course. There are left libertarians and rightwing libertarians in politics and economics - the leftwing ones emphasize civil liberties, the rightwing ones emphase economic laissez faire, with each man for himself and the devil take the hindmost. Occasionally left, center, and rightwing libertarians may agree on something - for example, if a government uses eminent domain to seize property of an individual, in order to re-sell the property to a big developer with the intent to bring in more tax money. As a left libertarian I find that a decent rightwing libertarian is likely to also oppose that - which I think was upheld by the USSC a few years ago.
In conversing with die-hard conservatives, I've found that they often (misrep) resent moderates and left wingers not as much because of policies, but simply because it is their policy. Among their opponents, the tendency is more that the right is criticized for its actions. Truth be told, both parties have more than their share of shortcomings and less than their share of ethics and competence.
Ironically, the left component of the SCOTUS upheld eminent domain "for the economic good of the community" and the right component opposed it. One would think that it would have gone the other way, as prior to the court ruling it was implemented by Texas Gov. Bill Clements for the Carlyle Group in an extremely questionable manner. I wonder if the SCOTUS deliberately overcompensates some decisions to make themselves appear non-partisan.