  There exists two types of semi-direct products between a Lie group $G$ and a
vector space $V$.
  The left semi-direct product, $G \ltimes V$, can be constructed when $G$ is
equipped with a left action on $V$.
  Similarly, the right semi-direct product, $G \rtimes V$, can be constructed
when $G$ is equipped with a right action on $V$.
  In this paper, we will construct a new type of semi-direct product, $G \Join
V$, which can be seen as the `sum' of a right and left semi-direct product.
  We then parallel existing semi-direct product Euler-Poincar\'{e} theory.
  We find that the group multiplication, the Lie bracket, and the diamond
operator can each be seen as a sum of the associated concepts in right and left
semi-direct product theory.
  Finally, we conclude with a toy example and the group of 2-jets of
diffeomorphisms above a fixed point.
  This final example has potential use in the creation of particle methods for
problems on diffeomorphism groups.
