Left and Right don't really show the complexity of politics anywhere. The Democratic Party of the past found it's birth in land holding policy and support of property owners and western expansion in the 1820s along side opposition to central banking. This Party drew it's support from the South becoming the political backing of Slavery and generally anti tariff because cheap foreign goods are good for wealthy southerners and bad for northern factory owners. Slavery was also bad for northern factory owners, and on the back of the Abolition movement the Republican Party was born. The North went Republican and then proceeded to hold the Presidency for most of the duration of the century. Having won the civil war the North and the Republican Party gained hegemony over much of American politics. They imposed heavy tariffs and generally backed the growth of corporations and later monopolies. In the South that still voted Democrat after the civil, race became a nucleus in politics to which the Republicans turned a blind eye and left the arena of race politics to the South. During this time America goes through a boom and bust cycle highly tied to corporate profiteering banking scandals and the highly volatility of an all gold standard currency. The Democrats meanwhile started getting involved with labor disputes in the coal mining states of the Appalachians where they became associated with the struggle for Unionization. At the time the battle was seen as the common southerner against the evil northern carpet bagger profiteering on the South. This along with Democratic party's stance on opening up the currency to a dual standard of currency backed by both silver and gold pushed the party line toward progressive politics. Meanwhile the Republicans enjoyed ever growing political power from the massive flood of corporate money into their coffers and enjoyed wide support still in the north but began to have it's progressive wing demand ever more for change. This led the Republicans to try and bury the career of Theodore Roosevelt who was the poster child of the progressive wing of the party by making him vice president which by the time was seen as a death knell to political careers. McKinley gets shot Theodore becomes president, reforms get enacted, and this only adds fuel to the progressive movement. Theodore doesn't run again and the party tries to reverse policy back to normal. This causes the progressives to revolt and try to push a third party ticket with Theodore again on the ticket. By this time the Democratic Party had reorganized with it's own progressive ticket and ran Woodrow Wilson, who then wins over the divided Republican Party. The new Democratic Party still has affiliation with race politics in the South (The party was heavily associated with the KKK) and still relies on it as it's base but now has gained traction in the North with progressives and pro-Union activists.

The Democrats pursue lower tariffs, recognize Unions, and bust Trusts. The Republicans though regain support following the economic boom after World War I and have a series of what are retrospectively some of the most corrupt presidencies in American history until that point barring Taft and McKinley. When the great depression hits the Republican Party and their dirty politics and inability to confront the crises get all the blame for the crises and they get dramatically swept out of power. FDR gets the presidency which sees massive intervention by the government in social programs and involvement in the economy which are now seen to be the hallmarks of the left. Post World War II the Republicans re-organize and grow in influence with the fear of communism on the rise and start pushing now hallmark conservative issues that align them against "liberals corrupting the American Way"(Heavily paraphrased). Economic stagnation in the late 50's sees the Republican on the out and out as the communist witch hunt ends. By this time the stigma of Republicans being carpet baggers is fading and a new generation of southern Republicans are getting into office who favor an America associated with traditional values and are god fearing anti-communists. Meanwhile the courts actually began to rule for the 13th Amendment and de-seggreagation begins, first the Army then schools, so on and so on. Northern Democrats tend to favor civil rights and the traditional republican base in the north is eroding so Johnson goes all in on backing civil rights with big grand legislative efforts. This garners him enough support to get elected but the Republicans of the South now become champions of the fight against civil rights, while the Republican party doesn't own it they choose not to comment. While the Dems take a hard line against anti-civil rights in their party line and distance themselves from party members that still support southern race politics. Then immedietly after civil rights the Republicans get delivered the perfect enemy: hippies. They are lazy liberal communist ungrateful un-Americans. All during a war. The youth backed the democrats due to the highly favorable opinions of their parents toward both FDR and Kennedy, along side their more liberal social policies and perceived youthfulness. And there you have it that is how the parties completely switched sides of the country. I hope I was able to cover the most important points of how this happened

There are a lot of details missing