Definitions: 
Mule (or muling) - 1. the act of combining dies that were never meant to be together. 2. the coin that results from the inappropriate combination of two different dies.

Intentional Mule - a deliberately created mule.  Examples include various Pattern issues and Restrikes made inside and outside the Mint.

Emergency Mule - a mule created out of necessity.  Examples include the 1795 "Heraldic Eagle" and 1798 "Small Eagle" Half Eagle that, because of the scarcity of dies in 1798, used old, left-over reverse dies from a style that had been replaced a year (or more) earlier.  Only a few examples of these exist in the U.S. series, most occurring when the Mint was in its infancy.

Error Mule - an accidental mule.  Because of the high level of quality control at the Mints, this is an inherently rare class.

Double-denomination Mule - an intentional or error mule that combines dies from two different denominations.  The rarest type in this class is the error mule that combines the obverse of a Washington 50 States Quarter Dollar with the reverse of a Sacajawea One Dollar (illustrated above).  This is the first instance in the entire history of the United States Mints that a coin has been created with different denominations stated on each side.

Rarity:
Only three different error mules are known in American numismatics.  The first mule is a Philippines coin, struck at the San Francisco Mint in 1918, that combines the obverse of 5 Centavos with the reverse of a 20 Centavos.  The second mule is a newly discovered coin that combines the obverse of a 1999 Cent with the reverse of a Roosevelt Dime (this piece was struck on the normal Cent planchet).  The third mule combines the obverse of a 50 States Washington Quarter Dollar with the reverse of a Sacajawea Dollar (all examples known were struck on the normal "golden" Sacajawea Dollar planchet).

Images courtesy of Heritage Numismatic Auctions

Known error mules:
1. Philippines coin, struck at the San Francisco Mint in 1918, that combines the obverse of 5 Centavos with the reverse of a 20 Centavos.  Rare.

2. 1995 Cent with the reverse of a Roosevelt Dime (struck on a normal Dime planchet).  Unique.

3. 1999 Cent with the reverse of a Roosevelt Dime (struck on a normal Cent planchet).  Unique.

4. Obverse of a 50 States Washington Quarter Dollar with the reverse of a Sacajawea Dollar (all examples known were struck on the normal "golden" Sacajawea Dollar planchet).  Seven known.

5. c. 1965 Two-Tailed Washington Quarter Dollar

6. 1859 Double-Headed Indian Cent

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