Many people use a modern misconception of the Left-Right dichotomy that says Left = More State Intervention, while Right = Less Government. This is not how Left-Right was originally conceived, or is used in the academic sense.
Rather, the more historically accurate conception is that Left = Egalitarianism is Justified, while Right = Hierarchy is Justified.
Thus, someone on the Left believes that all people are created equal and society should reflect this. While someone on the Right believes that some people are better than or superior to others and that society should reflect this (this doesn't mean necessarily that the better are to be treated better though, the concept of Noblesse Oblige, of the strong helping the weak is often seen here).
Note that interestingly, both concepts of Left-Right can be combined into a two-dimensional grid, with one dimension being Preferred Social Arrangement: Equality vs. Hierarchy, and the other being Agreeableness to State Intervention: Statist vs. Moderation. This can be used to actually better define the labels we often use, like so:
Liberal - Equality, Moderation
Conservative - Hierarchy, Moderation
Socialist - Equality, Statist
Fascist - Hierarchy, Statist
(Note: I use Socialism here as the extreme concept of socialism, that is with complete state control of the economy. I do not use Communism, because all "Communist" states were really socialist, and genuine communism has never been implemented outside of a few monasteries and the Israeli Kibbutz movement.)
Thus while both Liberals and Socialists agree that Equality is the goal, Liberals are more weary of the use of the state, while Socialists want to use the state to solve all problems relating to inequality. Similarly, Conservatives and Fascists agree that society should be hierarchical (note that different kinds of conservatives will disagree on the particular criteria, for instance most modern conservatives believe the hierarchy should be purely merit-based, while the old WWII fascists were a mix of merit, class, and racial supremacy-based).
Note also that both Liberals and Conservatives are much more weary of using government coercion to enforce their ideal, while both Socialists and Fascists see the state as simply the quickest means to their end. Note also that the Statist-Moderation dimension can also be described as the Authoritarian-Democratic dimension (note that I consider markets to be a form of economic democracy). As both Socialists and Fascists are more prone to Authoritarian extremism, while both Liberal and Conservative ideals are more democratically pluralistic.
This also means that the dimensions are continual. Thus, democratic socialism, fits somewhere between Liberal and Socialist, while both neoconservatism and Islamic conservatism (they're both cultural supremacy ideologies at heart) are somewhere between Conservatism (which could also be called traditional conservatism), and Fascism. Meanwhile, Centrism is between Liberal and Conservative, while most authoritarian dictatorships probably fall between Socialism and Fascism.