The ideological "sort" of conservatives into the Republican party and liberals into the Democratic party is a recent phenomenon. As recently as the late 1970s, the Republican party wasn't even clear on its stance on abortion. The sort was set in motion after Reagan was elected and wasn't really completed until after the Contract With America years.
There was a long time of Democrat domination of the southern states' politics from the end of Reconstruction that didn't end until about sixteen years ago. It's hard to assign ideological values like left/right, or even positions of, e.g., capitalism vs. not-capitalism, to the parties before the 70s, and even since (with some exceptions).
The Democratic party had something of a flirtation with the left side of the spectrum in FDR, but it wasn't anything like a defining characteristic of the party back then, and JFK went pretty much in the other direction. The Republican party had Presidents like Coolidge, who would definitely be on the right side of the spectrum, but also Nixon, who gave us price controls and proposed single-payer health care.
It's all a bit of a mishmash with a few salient exponents of somewhat-consistent ideology here and there, like Presidents Teddy Roosevelt (a progressive of sorts), FDR, Reagan, and now Trump. Things do seem to be clarifying in recent times, but even so, it's still not always all very clear. The fact is that the individual candidates have more going for or against them than just party affiliation or ideology.
There's an axis of populist economic policies in there, too; in the late 19th century this was strongly associated with Democrats, while Republicans pushed harder for "pure" laissez-faire capitalism.
That one is a bit more interesting, since in the modern day there are populist-economic wings of both major parties, though with very different approaches to implementing their ideas, surrounding a more laissez-faire "center".
