The problem is that people continue to use words like "left" and "liberal" as a vague proxy for party platforms, but it was never a one dimensional axis like that.
For example, twenty years ago you would expect to find Democrats arguing for "liberal" values like free speech and due process, and Republicans arguing for globalist foreign policy and military interventionism. Now it's going the other way. But which party is captured by the teachers unions and which party is captured by the oil companies hasn't changed, because it's not a one dimensional problem space.
That wasn't quite the same thing though. The impetus for the so-called flip was the Civil Rights Act.
It was becoming increasingly obvious that the Republicans were going to pass the Civil Rights Act and the racists were on the wrong side of history. But the racists made a final push to put it off and managed to put the Democrats in the majority.
Then the Northern Democrats voted with the Republicans to pass the Civil Rights Act anyway. The racists were livid. Their party betrayed them.
Nixon (yes, that Nixon) realized that it made the South his for the taking, so he took it. It was a realignment.
The Democrats like to portray this as the racists switching parties, but it was really the process of the racists losing and dying out. In 1880 the Democrats were the party of slavery and Jim Crow. In 1980 there was no party of slavery and Jim Crow. They lost.
That was the point when racism dissolved into classism. It's why Democrats insist on calling classism "structural racism" -- they've convinced people the "racists" are cardboard Republicans from the South, even though they're the ones tying schools to housing and restricting multi-family zoning in blue cities.
Not to say that the Republicans are saints. War on Drugs has been a predominantly Republican dung fire, for example.
It's not totally clear to me how you're using some of these words and their meanings can change a lot depending on context.
Like I would consider both the republican and democrat parties to be conservative, in the sense of generally resisting change, and liberal, in the sense of generally believing in market-based solutions to most problems.
The degree to which they're devoted to those things differs, but I think what differentiates them from each other is not their stances on these items.
"Liberalism is a political and moral philosophy based on liberty, consent of the governed and equality before the law. Liberals espouse a wide array of views depending on their understanding of these principles, but they generally support free markets, free trade, limited government, individual rights (including civil rights and human rights), capitalism, democracy, secularism, gender equality, racial equality, internationalism, freedom of speech, freedom of the press and freedom of religion."
I would say the vast majority of these principles are more associated with today's Republican party than today's Democrat party. I think two that might trip people up are "gender equality," and "racial equality," because the Democrats are the ones typically arguing for things approximating reparative justice/affirmative action/quotas, etc, but that is rather classified as "equity" rather than "equality." Equality, in the liberal sense, means equality before the law. Obviously reparative justice/affirmative action/quotas are the near opposite of equality before the law.
Internationalism and free trade are the two remaining principles that I wouldn't associate or disassociate with Republicans or Democrats, it basically just depends on what faction within the party. Bernie Sanders and Donald Trump are both against free trade, and more interested in prioritizing national workers over international trade. You also have free trade globalist types in either party.
Democratic party, obviously, is not remotely liberal, but that has been the case for some time. That doesn't mean good or bad, just not liberal.