Libertarianism can be described as: socially liberal, and economically conservative. I describe myself this way quite often.
In essence, however, it is political philosophy entirely independent of either conservatism or liberalism. The Libertarian Party itself was created as an educational venture, not as a serious political contender. It became bigger and more influential than anyone believed it could have, especially with Ed Clark's bid in 1980. Since that time it has suffered from various splits and fractures, and is seen less viable as a practical party, but it definitely has been accomplishing its original intent, which was to educate and inform the American public.
Libertarianism finds uneasy homes in both the Democrat and Republican parties, but this is at least a great start. The libertarian political philosophy has the ability to transform either party, but perhaps finds more acceptance amongst conservatives (or maybe more effort has been put into educating conservatives), and for that reason I think it would be easier to have an impact in the Republican Party. I think we've already seen this start to happen in a big way.
I support the Republican Liberty Caucus wholeheartedly, although I'm not yet a member (I only recently registered Republican!). I know little of the Democratic Freedom Caucus other than it was founded by ex LP members.
