CLs despise a large state regardless of who runs it and for what purpose. Reactionaries don’t long for a government that can be drowned in the bathtub, but they do hate the portions of it that they think work against the interests of “real Americans.”
The point of commonality is the desire to make large cuts in discretionary spending on the sectors of government both factions dislike, such as education. Expect that to be a major theme of the Trump years virtually from his first day in office.