  We give a combinatorial description of the ``$D_{2n}$ planar algebra,'' by
generators and relations. We explain how the generator interacts with the
Temperley-Lieb braiding. This shows the previously known braiding on the even
part extends to a `braiding up to sign' on the entire planar algebra.
  We give a direct proof that our relations are consistent (using this
`braiding up to sign'), give a complete description of the associated tensor
category and principal graph, and show that the planar algebra is positive
definite. These facts allow us to identify our combinatorial construction with
the standard invariant of the subfactor $D_{2n}$.
