  Two popular and often applied methods to obtain two-dimensional point sets
with the optimal order of $L_p$ discrepancy are digit scrambling and
symmetrization. In this paper we combine these two techniques and symmetrize
$b$-adic Hammersley point sets scrambled with arbitrary permutations. It is
already known that these modifications indeed assure that the $L_p$ discrepancy
is of optimal order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$ for $p\in
[1,\infty)$ in contrast to the classical Hammersley point set. We prove an
exact formula for the $L_2$ discrepancy of these point sets for special
permutations. We also present the permutations which lead to the lowest $L_2$
discrepancy for every base $b\in\{2,\dots,27\}$ by employing computer search
algorithms.
