Notes:
A single example of an 1859 Double-Headed Indian Head Cent is known.  Experts believe that additional examples may exist because of the way the coin was struck.  

According to Chris Pilliod, a blank die expected to be used as an "anvil" die (the reverse of the coin and the bottom die in a coining press) was accidentally fabricated as an obverse die.  A small portion of the side of dies are ground flat, ensuring proper alignment in the coining press.  Die-sinkers also use this "key" to properly align the dies for their purposes.  Thus, a coin struck using an anvil die engraved with an obverse design would be expected to have a medal alignment, which is indeed the case with this 1859 Double-Headed Cent.

Additional examples may exist because this coin was produced as a normal business strike under normal (albeit exceptional) conditions.

Die alignment:
Medal (approximately 170% off of the normal alignment for coins)

Die characteristics:
Side 1 - left edges of the 1 of the date pointing to the right half of the denticle below.  Broken R hub.  Raised die dot between the B and E of LIBERTY.

Side 2 - left edge of the 1 of the date pointing to the center of the denticle below.  Unbroken R hub.  Raised die dot on the temple area just left of the earlobe.  File mark.

Sources and/or recommended reading:
"Researcher Says Two-Headed 1859 Indian Head Cent Not Mule", COIN WORLD, March 5, 2001