In many European countries, liberals used to be left in the 19th century, whereas conservatives were right. Then liberals won, replaced conservatives on the right with socialists of various kinds on the left. In many European countries (not all), conservatism is dead, but the US, conservatism never died. Possibly because slavery required a hierarchical view of society; one of the tenets of conservatism, and at odds with the fundamental principles of liberalism.

But liberalism is primarily about freedom, about equality under the law, but not necessarily economic equality. Low taxes, little social security, wealthy industrialists owning everything is still compatible with equality under the law. So there are economically right-wing and left-wing interpretations of liberalism.

Similarly, people who are for or against economic equality can have liberal or authoritarian views on how to accomplish that.