I don’t find it fully satisfactory or convincing, but there’s an interesting essay from Rothbard (Left and Right) concerning this. He wrote that a key difference between the conservatives versus the classical liberals and the socialists that diverged from them, (which did not necessarily make these socialists polar opposites of the liberals on any kind of spectrum), was that while liberals sought liberal ends by liberal means, the socialists sought liberal ends by conservative means.
Conservatism is defined here in the historical context as one in favor of the absolutism, serfdom and hierarchy the old ruling class needed to retain power. It was a reactionary movement against the developments leading to the Industrial Revolution.
The socialists, like the liberals, accepted the new order of scientific progress, reason and advancement of the living standard of mankind as a whole (which the conservatives opposed), but unlike them, they embraced the State and the top-down organization that the conservatives favored, and opposed the revolutionary economic freedom and capitalism that was bitterly hated by the ruling class and that so utterly destroyed the Old Order of the conservatives.
They believed that taking control of the monopoly institutions, the power structure of the conservatives, with the right enlightened men in charge, would realize their liberal ends: the State can and ought to plan and organize society, if not realize that it is society and that society will inevitably reach an ideal state of social evolution (Social Darwinism).
The socialists then were the offspring of the liberals. It could be argued that among socialists there were left and right branches. Saint-Simon was among the right wing which was openly authoritarian and “a projection of conservatism trying to accept and dominate the new industrial civilization” in Rothbard’s words. See Bismarck.
Marx was representing the ‘libertarian left’ of the socialists, in that he sought a sort of third way where he believed the instruments of State power as they were understood (protection of the ruling class) could be used against itself and that the old order of economic exploitation would ‘wither away’. Yet the monopoly powers of the old power structure (economic planning) would be preserved under a new dictatorship of the proletariat in an interim phase as society ‘evolved’ into the final ideal stage of communism.
What defines the ‘left’ as we know it today then is its desire for liberal ends by enforcement of an aggressive monopoly. Their means for change and strategy to realize their liberal ends is essentially political, not economic. They want to change society by embracing and then transforming the power structure and the status quo.