Brian Kalt, a law professor and former college classmate of mine, has developed his own law of presidential facial hair:
I thought you might be interested in the following ironclad law of American presidential politics. I call it Kalt’s Law: “Under the modern two-party system, if a candidate has facial hair, the Republican always has as much, or more, than the Democrat.”
1. It would be the case that the Republican always has more, but for 1904 in which both the Republican Roosevelt and the Democrat Parker had mustaches.
2. This goes by coverage, not amount of hair as such (e.g., 1872: a full but short beard (Grant) is more than a long “neck beard” (Greeley). In 1904, Parker had a longer mustache than Roosevelt, but it doesn’t look like it covered more (see #1).
3. Here is a full table of pairings, Republican vs. Democrat:
1856, Frémont (beard) and Buchanan (nothing)
[in 1860, Lincoln didn’t grow his beard until after the election]
1864, Lincoln (no-mustache beard) and McClellan (mustache)
1868, Grant (beard) and Seymour (neck beard)
1872, Grant (beard) and Greeley (neck beard)
1876, Hayes (beard) and Tilden (nothing)
1880, Garfield (beard) and Hancock (vandyke)
1884, Blaine (beard) and Cleveland (mustache)
1888, Harrison (beard) and Cleveland (mustache)
1892, Harrison (beard) and Cleveland (mustache)
[1896 and 1900 featured no facial hair]
1904, Roosevelt (mustache) and Parker (mustache)
1908, Taft (mustache) and Bryan (nothing)
1912, Taft (mustache) and Wilson (nothing)
1916, Hughes (beard) and Wilson (nothing)
[1920 through 1940 featured no facial hair]
1944, Dewey (mustache) and Roosevelt (nothing)
1948, Dewey (mustache) and Truman (nothing)
[1952 through 2008 featured no facial hair]
4. No Democrat ever had a full beard.
5. Starting with their very first candidate, Frémont (and overlooking Lincoln’s quickly rectified lack of a beard in 1860), every Republican until 1896 had a beard. Besides McKinley, every one of them until 1920 had some facial hair.
6. Why would this be? Some amount of coincidence, surely. And some playing fast and loose with the terms (see 1904). But there may have been some difference in regional styles — other than Roosevelt and Dewey, the Republicans were all from the relative hinterlands (California, Illinois, Ohio, Indiana).
Note the important caveat that no candidate since 1952 has had facial hair (unless you count Nixon's 5 o'clock shadow, which of course would confirm the theory.)
Another caveat that may effect your view of Kalt's Law, for good or ill, is that Kalt is the person referenced at the bottom of this item.