  The process of symmetrization is often used to construct point sets with low
$L_p$ discrepancy. In the current work we apply this method to the shifted
Hammersley point set. It is known that for every shift this symmetrized point
set achieves an $L_p$ discrepancy of order
$\mathcal{O}\left(\sqrt{\log{N}}/N\right)$ for $p\in [1,\infty)$, which is best
possible in the sense of results by Roth, Schmidt and Hal\'asz. In this paper
we present an exact formula for the $L_2$ discrepancy of the symmetrized
Hammersley point set, which shows in particular that it is independent of the
choice for the shift.
