Before I go forward, let me give you a quick Too Long, Didn't Read: The premise of the question is wrong, but your question's premise is wrong for the right reasons.
So, the terms "Left, Right, Middle/Center" are used to describe politics in a regional context. If you look up the definition, American Right does not have much in common with European Right. An American Republican and a European Republican will want different things. A Liberal will be different to different people. A good deal of the politics arguments stems from a lack of a common frame of reference to what the terms mean. And America is a whole monkey wrench in and of itself.
So first, let's look at the Left-Right paradigm:
The term "Left" derives from the politics of the representative figures who sat on the left side of the French Parliament from the perspective of facing the chair, who were generally in favor of social equity, democracy, egalitarianism, liberalism, progressivism, and a slue of ideas that this brought forth (like socialism, which wasn't a thing back then, but is now).
The term "Right" came next from the opposition, who sat on the left side of the French Parliament. They were authoritarian, Royalists, reactionary, supported social hierarchies (aristocracy), institutions like the Church and the Crown, Nationalism, and Conservatism. Again, other similar ideologies are present (Facism, which didn't exist at this point).
Centralism/Middle-Aisle/Third-Way is harder to pin down to any ideology, but typically they would be all about compromise of the ideologies and trying to take the good points of both sides and get rid of the bad points. Some times this works (like American Liberitarianism, which is left on social issues and right on economic issues), and this is biased because I do like the American Libertarian ideology, so take that with a grain of salt. Some compromises are very bad ideas, a self-described Third-Way German representative took the Nationalist views that he loved on the right and some of the socialist views he loved from the Left and gave the world Nazism, which I'm pretty sure most people on Stack exchange would say American Libertarianism is way better ideology than Nazism. Also... center and middle are not related to seating beyond that if you're going to compromise, it is less physical work to sit in the middle... but the physical middle of most legislatures is reserved for party leadership.
So... in the terms that the Democrats are Left on of the chair and the Republicans are seated on the right of the Chair and thus the Party of the Right... but this is an ideology question.
It's from here that I have to stress, I will be talking about these terms in the most neutral terms possible. I'm quite aware of my biases and I'm going to try and restrain my comments to places I can point to in Wikipedia and reduce my opinion. I am going to give all ideologies a fair shake and will definitely note my opinion when I say things like "Nazism sucks (my opinion)". But I need to say this because I will be saying some things that may seem too nice to people who disagree with the ideology... or too harsh to people who agree.
Now, the reason why the Democrats seem to be to the right of Republicans on some issues and to the left on others is actually due to a dirty little secret. See, if you're not going to look at the definitons of the list of left and right beliefs, you could be forgiven for thinking I repeated myself... but with exception to listing royalists a few times... I did not repeat myself... those terms do mean different things and there are some cross germination ideas policies and sub-schools that narrow some pretty wide beliefs and others that marry them.
For example, a liberal can also be an authoritarian. In fact, John Locke's ideal liberalism was headed by an authoritarian but liberal absolute monarch. Monarchies are not always authoritarians, but Monarchies are never Republicans but Republicans can be Authoritarians (Soviet Union, Nazi Germany)... except in America where Republican means something else. Not all Democracies are Liberal and Monarchies can be Democratic (England and Japan). Almost all Liberal Democracies are Constititional, but exactly two are not (England and New Zealand) and Constitutionalism reject some core tenants of Parliamentarians but not all of them (never the less, the rejection was one of two big tent ideas that caused the Revolutionary War of the United States. A presidential government and parliamentarian government cannot be in one country but they aren't opposed to one another... those that try to get semi-presidential which has no hard rules other than they tried to do both. And similarly, a Federalist a Confederalist, and a Devolutionist all have differences between them but all oppose a Unitarialist but are fine with Unitarians (Okay, I made up Unitarialist as a word for people who support a unitary state. Unitarians follow a specific religious movement). All Communists are Socialists but not all Socialists are Communist (except in America, where they're all dirty filthy pinkos... that's an opinion... America has historically hard core rejected both ideologies). And Democracy is compatible with Communism (which actually loves Democracy), Progressiveness, and Liberalism, but a Democratic Progressive Liberal Communist simply cannot be. Two of the four terms are wholly and completely incompatible. Oh... and Conservatism is compatible with Every single one of these ideals name dropped in this paragraph with exception to one and only one ideal.
So... that's a lot of stuff to process to explain why Democrats some times seem more to the Right than Republicans and sometimes don't. And it has nothing to do with the Southern Strategy, though it's often ascribed to this.
The dirty little secret of the American Left-Right divide is... the American Right has more in common than the with the European Left than they ever had with the European Right. America was founded on major ideological consensuses of ideologies. They had a romantic love of Greco-Roman Democracy and Republicanism (which as they understood it, was not a Pure or Direct Democracy, but Democracy with limits on it... or what the rest of the world calls Representative Democracy, which at the time the United States implemented it, there was no term for the concept... so they called it Republicanism because they though Rome had it during the Republic). They had a problem with the Whiggory elements of Parlemenatrianism (i.e. Whiggorism was the concept that Parliment was the ultimate law of the land. Constitutionalism specifically opposes this as it holds that the final say in the law should be well documented and difficult to change by any one session of the Legislature and put hard limits on what the government can and cannot do).
But the most fundamental ideal was Liberalism which was a rather new school of thought that Americans absolutely loved as Liberalism was very pro-individual, pro-small government, and pro-limited government, all situations that worked well in America because the seat of government was one Atlantic ocean away from them and even to this day, much of the nations is very rural and largely not developed. The crux of all Liberalism is that the purpose of the state is to protect the citizens rights to "Life, Liberty, and Property." And if that phrase sounds familiar to you, well, there's a good reason why it should. Thomas Jefferson wasn't just plagiarizing Locke. Locke's summation was very well known, but Jefferson's paraphrasing added a twist from his own philosophy that made the statement the memorable "Life, Liberty, and the Pursuit of Happiness" that we know it as today. Jefferson subscribed to a simple Epicurian Philosphy, which aside from being a great line in a 1975's song, is an ideology that an individuals chief concern should be his or her own happiness, and the best judge of what will make you happy, is yourself. This also created a sort of idea of the Jeffersonian Democracy or Neo-Classical Liberalism, which holds that government should only be dedicated to protecting citizen's rights and nothing else, thus it should be small as this limits its ability to regulate societal ills that people don't like but that do not actually cause harm to other people. It also was very pro-economic liberalism, which favors deregualtion and less taxes (dropping Locke's Property was also a sneak from Jefferson and Ben Franklin to all property to be taxed by the government... because it was not something that expressly should be protected... you could still have it, but you need to pay to the government to procure it. They were not anti-tax... just anti-massive tax that the governed did not want but Parliament said they had too and cannot beat parliament.). And when I say that Democracy is compatible with Progressivism, Communism, and Liberalism, its the later two that don't get along well together. Liberalism is inherently capitalist and Communism is by definition, opposed to Capitalism.
Basically, you're on the right track with your assesment that there's something odd with the Left and Right devide... but it's mostly because American politics by and large rejected European Right wing politics and kinda have their own. Basically, America is a nation that has become so devoted to one key political ideology, it's like watching "Back to the Future" with two physics nerds who are engaged in a heated and pitched conversation about whether the Flux Capacitor fluxes capacitance or capacitates flux... to the point that they've lost site of anything about the nature of time travel. It's not wrong to have this devotion. It can be ammusing... but it is really quite weird and will result in some odd incongruities.
There might be edits... but this is long and kinda getting to a ramble and I need to end it for the time being.