So first thing I'd like to address is your misconception that most Americans are either Democrats or Republicans or the small subset of Green Party members. Actual polling typically finds that, when asked, the majority of U.S. Voters identify independent (no party affiliation) (over 1/3rd), followed by either Democrat or Republican (both slightly under 1/3rd), followed by the Libertarian Party and then then Green party (distant forth and fifth, but not enough that most polling will list them... they're pretty reliable to vote for the Republicans and Democrats respectively, though there are some pro-Democrat Libertarians). Because of this, in polls, the most important group to look too are the independents because they tend to be more willing to change their vote than non-independents. Underestimating them can and will cost candidates the elections, and in general elections, they tend to be the group that is courted.
While we're at this, we should get some political terms out of the way:
Left/Right - The right/left divide tends to be determined by national politics rather than globally defined. For example, a right wing Canadian PM would probably be at best a moderate pro-right person in the United States, while a left wing Japanese politician might be more close to a moderate pro-left person in the U.S. at best. The divide is generally regional and not broadly in lock step between nations. To give one real life example, Angela Merkle, the former Chancellor of Germany, is a member of the Christian Democratic Union of Germany (CDU) which is a center-right political party in Germany. However, she would have difficulty courting the Center Right in the United States, and her U.S. supporters tend to be on the U.S. political left. The terms got their name from the early French Revolution and the National Assembly. Those who supported the King would sit on the right side of the room (when viewed from the Assembly President's podium) while the revolutionaries sat on the left. Since then the right has been associated with conservative politics and the left has been concerned with reformist politics.
Meanwhile, in the United States, the Left/Right divide is less understood and generally falls into the Liberal/Conservative divide. This tends to be more of a false dichotomy as Liberalism and Conservatism are not opposing ideals. America is founded on the principles of Liberalism and it's generally the core beliefs for which the Revolutionary War was fought (Colonial Americans were avid readers of Liberal thinkers of their day and age. Today this is know as Classical Liberalism but at the time the ideology was too new to have large divides). As a general rule, Liberalism is the belief in the concept of "natural law" which holds that in the setting of the wilderness (with no other human contact), anything that people can do is something the government cannot take from them, and that the government must protect the rights of the people. Generally these rights are broadly summed up as "Life, Liberty (the freedom of movement), and property" (John Locke, who was one of the first Liberal philosophers, used the term property. Thomas Jefferson, when writing the Declaration of Independence, changed property to "happiness" because he realized there were people who weren't materialists and Jefferson was a hedonist, which is all about the pursuit of happiness... in fact, the Declaration of Independence is a very short summation of Classical Liberal philosophy as a whole). Liberalism tends to be highly individualistic and pro-capitalistic.
Conservatism is a less strictly defined political philosophy and generally can be summed up as "If it ain't broke, don't fix it." Conservatives tend to favor established institutions over reform and change and hold to the belief that the changes are at the root of the problems rather than the institutions being flawed. This doesn't mean that they are opposed to change, but that they need to be convinced that change is necessary or that the changes need to be less dramatic. They also do not favor any form of economics. Vladimir Putin is a conservative politician... but he is in favor of bringing back Soviet institutions and policies, not bringing further capitalist economics to Russia. It's a very regional based. In the United States, Conservatism tends to have liberal doctrines as it's planks (in fact, in Modern European politics, pro-Liberal parties tend to be classified as center-right).
In the United States, the two party system and first past the post nature have lead to both parties becoming "Big Tent" parties, where as in Europe you will have a handful of big parties that tend to make coalitions with the few smaller parties in a coalition (since this tends to determine the Prime Minister, a loan member of a fringe party can be a king maker... in the U.S. it's rare that the party with the majority fails to nominate a speaker from their own party, but it's harder for the majority to get something through Congress than it is for the minority party to stop it. The reason for this is the U.S. political system tends to let elected officials be less bound to party platform than other European systems. For example, Joe Manchin, a democrat, tends to vote against many democratic supported legislation because he represents the state of West Virginia, a very republican state, and wants to be re-elected... so it's in his interests to not vote with the Democrats on many things. One could say that Manchin's personal politics are unknown, since he's trying to keep his seat in a constituency that opposes his party quite often.
Historically, the longest standing political divides in the United States tend to be along the lives of an Urban/Rural divide, though who represented these divides has changed (For most of their history, the Republicans tended to be the Urban party and the Democrats tended to be the Rural party... though this has switched in the past 50-70 years. It also tends to fall on a professional/working class divide although this is less reliable than in the past. Even a Christian vs. Atheist divide is not reliable as Catholics tend to favor Democrats over Republicans (again, this is due to the Urban/Rural split. Catholicism tends to be a more Urban religion, while Protestants tend to be a more rural religion. As is Islam and Judaism, which tend to be much more pro-Democrat than Republican.).
Interestingly, both the Democrats and Republicans can trace their parties and ideologies back to the historical Democrat-Republican Party or more importantly, the Democratic-Republican Party founder, Thomas Jefferson. Democrats are directly descended by way of Andrew Jackson's influence on the party. The Republican party tended to be indirect in that it was founded by a coalition of members of the defunct Whig party and the defunct National Republican Party, mixed with several minor parties that supported abolition of slavery.
It should also be pointed out that for much of American political history, there tended to be a mix of ideologies in both parties, to such a degree that JFK was as staunch an anti-communist as Richard Nixon and Teddy Roosevelt went on to form the short lived Progressive Party (AKA The Bull Moose Party) and under the Sherman Anti-trust act, sued 45 companies. His successor, Republican Taft, sued 75 companies under the same act. Both Roosevelt and Nixon were also notable pro-environmental Presidents (with Roosevelt founding the National Parks system and Nixon forming the EPA). Even states reliability to a particular party are not absolute. California has an intersting history of being a long term swing state... that is, it will go for long stretches of being reliable for one party, only to switch to the other party for a lengthy period. From 1952-1992, California only voted for the Democratic Candidate for President twice (In 1964 and again in 1992). They went for the Republican in every other presidential election (a total of 9 elections. It should help that a Californian was on the ticket in 5 of them, was on the Republican Ticket for VP for two, and was the VP of a Californian in 2 more). During this same period, Texas has been a reliable Democrat stronghold.
All this is to say that American's are not evenly split on political ideologies as one might have you think. In fact, the easy fit dichotomy or U.S. politics is a myth that Republicans and Democrats will happily support because the more people who are exclusively party voters, the less they have to worry about the independents who are not loyal to either party.
And that said, independents are wild cards as many of them may favor one party over the other, but not to a point that they are reliable and will break for a spoiler candidate of someone from the other party (or just not vote!) if their preferred party candidate is off putting. As an independent voter, I voted for three separate parties' candidates before I voted for a party for a second time. And I'm 3/4 in terms of voting for losing candidates.