Bill Clinton (1993-2000) - two-term centrist Democrat in the middle of an otherwise Republican dominated era.  Elected with the help of third party candidate H. Ross Perot in 1992.

Go back exactly 40 years and you find:

Dwight Eisenhower (1953-1960) - two-term centrist Republican in the middle of an otherwise Democratic dominated era.  (No third party help in his case)

Exactly 40 years before that:

Woodrow Wilson (1913-1920) - two-term centrist Democrat in the middle of an otherwise Republican dominated era.  Elected with the help of third party candidate Theodore Roosevelt in 1912.

Other interesting facts:
Republican-dominated era: 1896-1932 (except Wilson)
Democratic-dominated era: 1932-1968 (except Eisenhower)
Republican-dominated era: 1968-? (except Carter and Clinton)
Doesn't quite follow the 40/80 year theory but it's pretty close.

2000 election:
Bush (R)  47.87% popular  271 electoral
Gore (D)  48.38% popular  266 electoral

40 years before that, the 1960 election:
Kennedy (D)  49.72% popular  303 electoral
Nixon (R)  49.55% popular  219 electoral

Both elections hotly contested and racked by charges of vote fraud.

1924. Democrats lost, amid the early stages of a resurgent liberal movement (also note that the most obvious sign of that movement - LaFollette's third party run - came 76 years before Nader played the same role in 2000.  Not quite 80 years but close.)  
      
1964.  Republicans lost, amid the early stages of a resurgent conservative movement.  
      
2004.  Democrats lost, amid the early stages of a resurgent liberal movement.

Also, the theory falls apart if you go back before 1896, so it's not meant to be any kind of scientific political analysis, just an observation.

There's an interesting book that sheds a lot of light on the current parallels with the 1920s, and also backs up and has a plausible explanation for my theory about politics running in 40/80 year cycles; "The Fourth Turning" by Neil Howe and William Strauss.