I’ve been delaying posting this for a while, but its time. Its a perhaps idiosyncratic view of where BHL fits into the set of political philosophies nearest to it (in some sense of “nearest”). Some notes follow the taxonomy.

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1. Liberalism: A family of theories that take liberty and equality to be the most important (and guiding) values in the organization of a just state. Historically, some have added fraternity to liberty and equality. Different liberals understand these core values in different ways.
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1.1. Libertarianism: A family of liberal views that take negative liberty (freedom from interference) to be the most important (and guiding) value in the organization of a just state, insisting it must be present for all. (Some call this “classical liberalism.”)
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1.1.1. “Right” Libertarians: A family of libertarian (and hence liberal) views that take property rights in external goods (often justified with appeal to self-ownership) to be absolute or nearly absolute. That emphasis sometimes makes them consider coercive redistribution to be necessarily immoral. Some (perhaps most) in this family may nonetheless agree that it is a benefit of the view that if instantiated, most would do better than if society was organized along other lines.
1.1.1.1. Minimal State Theorists or Minarchists: A family of “Right” Libertarian views that take government to be an element of the state that should be used only to protect negative liberty.
1.1.1.2. Anarchist Libertarians: A family of “Right” Libertarian views that believe negative liberty is better served with the absence of government than with its presence.
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1.1.2. BHLs: A family of libertarian (and hence liberal) views that also share a deep concern to prevent suffering (and perhaps promote at least minimal individual well-being). Some in this camp may approve of limited government interventions to end suffering; all agree that allowing individuals extensive (negative) liberty is likely to create the least suffering possible. Some may favor pretty strong, if not absolute property rights (the latter is more likely with 1.1.2.2 than 1.1.2.1).
1.1.2.1. Minimal State or Minarchist BHLs: A family of BHL views that take government to be an element of the state that should be used to protect individuals, especially their negative liberty, but sometimes also minimal levels of well-being.
1.1.2.2. Anarchist BHLs: A family of BHL views that believe individuals are better served (both in terms of negative liberty and well-being) with the absence of government then with its presence.
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1.1.3. “Left” Libertarians (“LLs”): A family of libertarian (and hence liberal) views that take liberty to be of such value that it must be distributed equally (and may believe the “Right” Libertarian way of defining and emphasizing property rights hinders that). Likely also to take equal opportunity of well-being as a value, but to consider it extensionally equivalent to equal liberty. Some in this camp limit property rights in natural resources, perhaps defining property (or the bundle of rights that it is) in such a way that makes this clear.
1.1.3.1. Minimal State or Minarchist LLs: A family of LL views that take government to be an element of the state that should be used to guarantee the equal distribution of liberty (and equal opportunity of well-being).
1.1.3.2. Anarchist LLs: A family of LL views that believe private agencies can better guarantee the equal distribution of liberty (and equal opportunity of well-being) than a government.
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1.2. Egalitarian Liberalism: A family of liberal views that take liberty and equality of well-being to be the most important (and guiding) values in the organization of a just state, with an emphasis on the latter enabling its proponents to sometimes limit liberty. Such theorists nonetheless take liberty to be centrally important.
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1.3. Communitarian Liberalism: A family of theories that take liberty, equality, and fraternity to be the most important (and guiding) values in the organization of a just state, with an emphasis on fraternity enabling its proponents to sometimes limit liberty–or equality.
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Note 1: In none of these did I offer any indication of how the particular view would be defended. This is important. Political philosophy is a part of moral philosophy, but political philosophies are not complete moral theories. Different political philosophers can and do use different moral theories to defend their views. There are historical and contemporary libertarians, to take our example, that are consequentialists, others that are deontologists, and others that are teleologists. Admittedly, this makes the whole terrain extremely variegated. Some think it an advantage to libertarianism that it can be defended by a variety of ethical theories, but I’m not confident the same isn’t true of other political views. The problem, there, is twofold: (a) determine the best moral theory and (b) determine which political philosophy that moral theory actually supports. Alternatively, one might seek to show that all contenders for best moral theory actually support the same political philosophy.

Note 2: I am not offering necessary or sufficient conditions in the taxonomy. I suspect an attempt at such would fail, but I’d be happy to see it.

Note 3: I use the “left” and “right” nomenclature though I dislike it.

Note 4: Thanks to Matt Zwolinski and Hillel Steiner for help formulating some of the above.