There is traditional conservatism that favors an authoritarian state (monarchy), a state religion and an established order as guardians of tradition and custom and bulwarks against innovation and change. This is called Traditionalism. It's chief opponent is Liberalism. Traditionalism is reactionary, and it regards Liberalism as radicalism.
The other type of conservatism is actually an aspect of Liberalism. This sort of conservatism tends to be economically liberal but politically and socially traditional. This is the conservative-liberal divide in US and UK politics for example.
Within the conservative aspect of Liberalism there are several approaches. One is neoliberalism and the other compassionate conservatism. This article explores this dichotomy in UK politics, specifically the differences in approaches of David Cameron and Theresa May.
While the Cameron government espoused the neoliberal principals of fiscal consolidation and austerity economically and globalism and open borders politically, the May government has jettisoned that approach in favor of a more inclusive capitalism, declaring the age of Thatcherism over in favor of nationalism and using the power of the state for public purpose including promoting the general welfare.
The post suggests that instead of abandoning of conservatism, this can be viewed as returning to the policy approach of Benjamin Disraeli's One Nation Toryism.
Longish, but worth the read. Although it is about the UK specifically, the issues are similar in the US. A good summary of conservatism within the Liberal political tradition.
Open Democracy
A battle for the meaning of British Conservatism
Ben WaiteA battle for the meaning of British Conservatism
