Is there a such thing as a 1960 Wheat Penny?

There should be no 1960 wheat cents, as the Lincoln Memorial reverse was used starting in 1959. In theory an old wheat cent reverse die could've been used with a 1960 obverse die, though I don't know of any that have been verified. If you have one, look for a seam around the edge or rim of the coin. Also examine carefully the date. If you cannot see any seam or any evidence of other alteration, get the coin certified.