There is no right-wing like the right wing in the US. I suspect that being on top of the world for 50 years would do that to any nation. :-(
The terms "left" and "right" appeared during the French Revolution of 1789 when members of the National Assembly divided into supporters of the king to the president's right and supporters of the revolution to his left. The terms then grew to mean (in French) that left-leaning parties are the "parties of movement", and right-leaning parties are "parties of order".
The term then got exported to English, and where people stand on "Right/Left" in the US now lines up with a few core sympathies:
Capital/Labor,
Religious/Secular,
Rural/Urban,
old white men/everyone else,
the last one being the most problematic at present.
Americans nowadays bag on Republicans because the Republican constituency has undergone a strange shift over the years. When it was first formed, the Republican party was the party of the free, working, white man. Throughout the century between the American Civil War and the Vietnam War, It used to be that the Republican party formed a useful and relatively stable coalition of rich white men, who were in political power all throughout the country. Because the Republican Party has historically been less supportive of the progress of civil rights (e.g., desegregation, equal pay/abortion, gay rights), the Republican constituency has become more narrow over time. By contrast, the Democratic party has been more successful in keeping up with US demographic changes.
But ever since about the 1990s, demographics have changed and the domestic economy has worsened, and the interests of the rural, the rich, the religious, and the racists have diverged somewhat. As a result, Republicans, despite enjoying a strong identity, have become as factionalized as the Democratic constituency legendarily was in the latter half of the 20th century, but the Republicans have less practice at the politics of maintaining large coalitions.
This gives rise to the modern embarrassments, where the Republicans, while on paper a majority party in the lower legislative house, are really only a majority coalition. They can only agree to block what the President from the opposing party does, but not on any coherent agenda beyond that. Meanwhile, extremists of all four major Republican types are trying to make it clear that their version alone represents the one true Republican party.