It's kind of silly to call the two major parties in the United States either left-wing or right-wing. Political Scientists have studied the "party cleavages" of the Republicans and Democrats in great detail. In comparison to other states' party cleavages, an ordinal scale is created: 1 being a statist, far-far-right government (Nazis), and 10 being a communist, totalitarian state (far left, Stalin).
Due to the nature of the two party system, in order to gain the most votes the parties in the United States gravitate towards the center. How much so? The democrats are rated at a 4.8 (just slightly below centrist, hardly enough to describe as left-leaning), and Republicans at a 5.5 (just slightly above centrist) -- meaning both parties are quite similar and have only minor differences. In the end, they are both centrist parties.
In the United States, there are authentic left-leaning parties like the Green Party, and right leaning ones like the Conservative Party; but these never get enough votes to pass thresholds for campaign financing nor seats in a federal body like Congress.
In conclusion, the major parties of the United States are both centrist, and while their rhetoric might illustrate contrasts between them, in reality they are very close to each other in the political spectrum.