Consider classical liberalism then. Originally considered as a movement of the left, is now considered extreme right.
Socially they are liberal, a facet of the left. Economically they are liberal, a facet of the right.
So what are they?
What qualifies as left or right?
Yes but we must use the present day spectrum. Definitions change over time. Back 'in the day', a leftist was simply a social agitator, and rightist a 'change nothing whatsoever' conservative. Clearly this spectrum has no use nowadays (and isn't used), since for example, this would make a maoist to be right wing in current day china (since he would be attempting to 'conserve' communism in china).
What specifically constitutes the right and left, and why these are polar opposites, has been something I have been pondering for some time. So far I have reached the following model:
The Right are those beliefs consistent with humans increasing their position. That is to say, 'pro-life' in the basic biological sense.
The Left are those beliefs consistent with humans decreasing their position. That is to say, 'pro-death' in the basic sense.
If you'd like me to detail why this is, I can, but the post will be long and non-NBN related
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The issue is that you are trying to define a group of beliefs according to 'left and right' - whereas this is impossible. An isolated belief can be characterized, for example 'offensive speech should be banned' is clearly Left, 'all property should be private' is clearly Right. However if a (confused) person holds both of these beliefs, he can only be graded on the spectrum.
Libertarians, for instance (which I believe is more or less the same as a classical liberal), hold beliefs on both sides (although mostly on the right). Regarding them being 'social left', not necessarily. Typically on immigration, left. On abortion, divided. On race/class/gender, right (no affirmative action, no quotas, no disparity targeting privileges). On social conventions, to be logically consistent with 'liberty' they are mostly left (gay marriage fine, incest fine, polygamy fine, public nudity fine etc). On freedom of speech and action, right (full free speech, all non-rights-violating actions fine).