  It is well known that the $L_p$-discrepancy for $p \in [1,\infty]$ of the van
der Corput sequence is of exact order of magnitude $O((\log N)/N)$. This
however is for $p \in (1,\infty)$ not best possible with respect to the lower
bounds according to Roth and Proinov. For the case $p=2$ it is well known that
the symmetrization trick due to Davenport leads to the optimal
$L_2$-discrepancy rate $O(\sqrt{\log N}/N)$ for the symmetrized van der Corput
sequence. In this note we show that this result holds for all $p \in
(1,\infty)$. The proof is based on an estimate of the Haar coefficients of the
corresponding local discrepancy and on the use of the Littlewood-Paley
inequality.
