It's pretty clear that the Tea Party is of the Right, and although the Tea Party sometimes makes claims to being libertarian, we need not make an equivalency between the two. Philosophically, does American-style libertarianism dovetail with conservatism or with liberalism? Pragmatically, do libertarians align themselves more with Republicans (i.e., the Right) or with Democrats (i.e., the Left)? A common argument from libertarians themselves is that they are neither Left nor Right; they proffer a Nolan chart to measure political views on two dimensions most relevant to them: economic deregulation vs. market regulation and personal freedom vs. authoritarianism.
Philosophically, many libertarians find roots in what they refer to as classical liberalism, a point of diversion with contemporary U.S. liberals. They feel that modern U.S. liberals abandoned their commitment to free markets and thus no longer deserve the name. Contemporary U.S. liberals have developed a subsequent intellectual tradition that in some ways brings them closer to European social democracy while emphasizing the welfare state's role in increasing individual freedom. Leftism, though, if seen as a tendency rather than a rigid set of political planks seems to be; leftism then can be seen as a commitment to level the playing field and removing unjust power. The Right would be the Left's opposition then, a reaction; the Right would attempt to justify the status quo and empower institutions that guard against societal change. In the 18th and 19th centuries, the old guard would have been the monarchy, the aristocracy, and the Church; today it might well be corporations.
Unrestrained free markets, though, definitely can bring about rapid change. In this sense, the market is not a conservative force. However, if the benefit to this change goes mostly to a select few with the risk falling on most people, is an unrestrained market necessarily a force for positive change? When competition ends up benefiting employees and consumers while providing desirable innovations, it's a boon; when it restricts freedom for most, it's a negative. In the latter case, large corporations sometimes use their clout and wealth anticompetitively, so it may be said to no longer be a free market.
Largely, I think it's what libertarians prioritize. Libertarians who are willing to sacrifice personal freedom for economic freedom cleave to the Right, and those who prioritize personal freedom tend more to the Left. Those who temperamentally align conservative and do not see a contradiction between their talk of freedom and their desire for a Christian nation definitely fall on the conservative Right.