Left + Right in modern discussion are non-descriptive words and as such tend to vary widely in definition. The classic definition comes from european parliaments. In previous centuries, when monarchy was far stronger and the established power, in those lands where a parliament or senate existed, those who wanted to abolish monarchy and establish democracy sat left. The monarchists who wanted to keep or strengthen the established power of the king sat right. Keep slavery on the right, establish universal suffrage left. Liberalism demanding basic human rights was a far left concept for a very long time, and authoritarianism a far right concept. But it is not all about progressives and conservatives.

To understand "Left+Right" ask the question: if we seat these people next to each other on thanksgiving (or in parliament) how likely is physical violence.

In the early 20th century in Germany, in the Weimar Republic to be precise, monarchy has been abolished and democacy established. The monarchists were still sitting at the right in parliament, and they wanted to abolish democracy and establish a kingdom. At the far right sat the fascists. The monarchists and fascists both were nationalistic, authoritarian and somewhat accepting of capitalism.

At the far left sat the communists, who were about international communist revolution and abolishing capitalism. The communists themselves were split in liberal communists and authoritarian communists. The later took power in Russia and later Stalin became their dictator, and the european communists faced a dumb choice between "international communism" and "liberal-democratic communism". The Stalinists sat far left and the those favoring liberal democratic communism sat between them and those favoring democratic capitalism with social benefits within the current nation. Then the far right abolished democracy in Germany and established a dictatorship.

In early 20th century "liberalism and democracy" became a center opinion, because both the far left and the far right were pushing for an authoritarian dictatorship, with the "moderate left" and "moderate right" typically agreeing with their extremists on other things while being more liberal-democratic.

But those times are long gone in europa and have never the USA has a bit of a problem with this eurocentric view anyway, because they haven't had a party sitting anywhere in parliament being in favor of abolishing their facade of a liberal democratic republic hiding corporate rule in centuries. At the moment many far right groups are trying to claim liberalism as their concept, as a "conservative" thing pointing out that progressive political streams like environmentalism have authoritzarian tendencies as they strongly regulate the economy. But in truth the parliamentarian right wing is not about conserving your right to destroy the environment, they care far more about conserving the established power of large corporations.