Looking for a book where the good guy lives backwards through time and the bad guy lives forward. The good guy was an alien and keeps getting killed by the bad guy. The bad guy was a Neanderthal(?) who keeps killing the good guy through time until the beginning of time which is at the end of the book.
This is the first book in the Orion series by Ben Bova.
The good guy is John O'Ryan aka Orion the hunter, and he is battling Ahriman aka the Dark Lord. As you say they move in opposite directions through time.
Ahriman is a Neanderthal. The conflict arises because the Neanderthals had to die out for the humans to live. At the end O'Ryan finds himself feeling that Ahriman's cause is basically just and he changes reality to allow the Neanderthals to live.