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Zaq comments on Can You Prove Two Particles Are Identical? - Less Wrong
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Sometimes the electron meets itself coming the other way and together it turns into a photon. And sometimes that photon can't decide whether to go forwards in time or backwards in time, so it does both -- but it doesn't always do so as an electron/positron. So really there's only one particle, period. ;-)
But there's also the observed matter-antimatter asymmetry. Observations strongly indicate that right now we have a lot more electrons than positrons. If it was just one electron going back and forth in time (and occasionally being a photon), we'd expect at most one extra electron.
Not to mention the fact that positrons = electrons going backwards in time only works if you ignore gravity.