This question speaks to the "start from the beginning" meaning, but in yesterday's USA Today, there was a headline about Obama and Romney starting from scratch because they were even in a poll. Have you heard it mean to be at a tie?
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Per other answers, USA Today's usage here is at the very least "questionable". In a sports context where the scores are equal some way through a match, you could reasonably say "It's as if they're starting from scratch". But in the context of electioneering, the "metaphor" doesn't work well. It gets mixed up with the idea that preceding campaign strategies might be replaced, since the expression start from scratch is...
In this and the related up to scratch (good enough to meet a particular standard), the word "scratch" comes from a boundary or starting point scratched on the ground in a sporting context. Originally it signified having no advantage or handicap (weaker contestants might be allowed to start beyond the scratch as a "head start"), from whence the meaning has shifted to the modern sense highlighted above (effectively, "gaining no advantage from preceding activities/knowledge").
The article doesn't appear to explain what they mean by the headline. As others have noted, the phrase means "start from nothing", but clearly the two men are not starting from nothing now: the article explicitly points out that each has a substantial base and that they are fighting over the undecideds.
After seeing the headline but before reading the article, I thought they were going to say something like, Both men's campaign strategies have proven ineffective, and so they must start over with new ideas or new plans. An argument like that would be consistent with saying that they are starting from scrath. But the article doesn't say anything like that.
In short, I think the headline has nothing to do with the article. It's bad editing.
A "real" start from scratch for a contest is 0-0.
But if it's a tie, say 45-45, the impact of the contest is basically the same as if you were starting from 0-0.
I consider this headline misleading because one person was actually ahead, 47-45 (that is, within the 3-point margin of error). I consider this a "stretch" on both counts, but the paper was using an analogy of "starting from scratch." I call it "journalistic license."