We've only had two elections where the dominant party was ousted from power, realigning elections (1824, 1844, 1876) not included. I suppose that's what you get when a bunch of internet leftists chart the course of history. 
I think a solution might be to merge Republicans and Democrats. Lincoln+Cleveland votes would have been more than enough to beat Stanton.
It seems to vary from TL to TL. Cath's first series had significant periods of conservative dominence, left vs. far-left conflict, swing elections, and mainstream liberal dominance (a good variety). The one run by Cath then Dallas then Alfred and I and now Spiral has generally seen more conservative dominance than anything else (albeit with some exceptions). The one run by Harry S. Truman has (so far) generally been dominated by the left (with some exceptions like the D-R era and the economic conservatism of anti-slavery Northerners like Hale).
We'll see who (if anyone) is in the driver's seat whenever I finally get around to starting my timeline/election series which will be a little different in the sense that I'm also going to include the full makeup of the Supreme Court in each GE thread meaning the voters' choices will effect who gets appointed, when there are vacancies, and the outcomes of major decisions. For example, depending on the makeup of the SC at that point in the TL, it'd be entirely possible for Brown vs. Board or Roe vs. Wade to have the opposite outcome of the RL case. I imagine this would make people sometimes vote with concerns about potential SC vacancies in mind, as is sometimes the case IRL.
But I digress, I voted merger. Should've gone with my gut and voted Stevenson
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