Old Presidents' Parties Today

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Reagan - Republican / Establishment conservative
Ford - Republican / Establishment moderate
Nixon - Republican / Establishment conservative
Johnson - Democrat / Establishment moderate
Kennedy - Democrat / Establishment moderate
Eisenhower - Republican / Establishment moderate
Truman - Democrat / Blue Dog Democrat
Roosevelt - Democrat / Establishment moderate
Hoover - Republican / Establishment moderate
Coolidge - Republican / Establishment conservative
Harding - Republican / Establishment moderate
Wilson - Democrat / Establishment moderate
Taft - Republican / Establishment moderate
Roosevelt - Republican / Establishment conservative
McKinley - Republican / Establishment moderate
 

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Are there any pre-1897 presidents from either party you see switching? Cleveland's obvious, though I wonder what your take on Harrison and Garfield might be.

Cleveland - Libertarian or Republican. Probably the former, given that most of the 'libertarian' wing of the GOP is in actuality paleoconservative and favorable to protectionism rather than embracing free trade. Cleveland would probably fit in pretty well with our modern MRAs, too, given that he didn't think that women were smart enough to exercise the franchise (curiously this seems to be a fairly consistent view from the 'classical liberals', iirc)
Harrison - Democrat, I'd say. Probably a protectionist liberal in the mold of Sherrod Brown, given his attempt at smashing Jim Crow in 1890 that unfortunately failed and his opposition to monopolies. Harrison was really our first progressive President in the modern sense, if you ask me.
Arthur - Republican, of a moderate establishment variety. The guy did after all oppose immigration restriction, but at the same time, he purged the civil service of blacks via 'Civil Service Reform'.
Hayes - Republican / Moderate establishmentarian
Grant - Democrat / Liberal establishmentarian (think Bill de Blasio or Liz Warren)
Lincoln - Democrat / Liberal establishmentarian, probably.
Buchanan - Republican / Conservative establishmentarian
Pierce - Republican / Conservative establishmentarian
Fillmore - Constitution or Tea Party Republican
Taylor - Republican / Moderate establishmentarian
Polk - Republican / Conservative establishmentarian
Tyler - Constitution or Tea Party Republican
Harrison - Republican / Moderate establishmentarian
Van Buren - Libertarian or conservative Democrat
Jackson - Democrat, probably a Blue Dog or populist Democrat
Adams - Republican / Liberal Northeastern Republican
Monroe - Democrat / Conservative Democrat
Madison - Democrat / Conservative Democrat
Jefferson - Green, left-wing Democrat, or Socialist, depending upon what period of his life we're talking about. Jefferson was always something of a political radical, he supported the Jacobins in France, etc.
Adams - Republican / Conservative establishmentarian
Washington - Republican / Conservative establishmentarian

Of course, were all these people resurrected today, probably everyone who served before 1860 would be a member of the Constitution or Libertarian Party, given how integral white supremacy was as a concept in the early Republic. 

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Cleveland and JFK would be GOP'rs today. In fact they'd be Tea Partiers. It's sad many on the left think JFK would stand with them. The guy was a fiscal hawk who believed in sound currency (unlike the left of today).

JFK would definitely be a Democrat, although he'd be a moderate or conservative one, as he was in the 1960s. I agree with you that a liberal or left-wing JFK is wishful thinking on the part of today's progressives, but there's no way he'd be a Republican. I could see him as a Cuomo-Clinton type Democrat, though.

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Not doing any President before the 20th Century

T. Roosevelt: Democratic
Taft: Republican
Wilson: Democratic
Harding: Republican
Coolidge: Republican
Hoover: Republican
FDR: Democratic
Truman: Democratic
Eisenhower: Democratic or Independent
JFK: Democratic
LBJ: Democratic
Nixon: Democratic
Ford: Democratic
After Carter would be the same 

Um... what? Nixon almost certainly would not be a Democrat.

If he had the political views he had in the 1970s, he'd be some sort of an authoritarian Lieberman-esque Democrat. Of course were Nixon alive to-day he might "adjust" some of his views as is the case in TNF's model.

Nixon was always a conservative. The fact that he signed off on some liberal legislation does not make him a moderate or a liberal. Arguing otherwise is like arguing that W was a liberal because he supported Medicare Part D and signed off on a minimum wage increase.

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There are many things that Theodore Roosevelt was, but progressive was not one of them. Roosevelt was an opportunist who tried to make the progressive movement his own in order to blunt its appeal and dilute its outcome. The best way to think of Roosevelt is an American Bismarck, given that he referred to the La Follette progressives as the 'lunatic fringe' and was less concerned with most progressive pet issues than he was with making sure the country didn't descend into anarchism or whatever the line was a century ago.