  We study the $L_p$ discrepancy of two-dimensional digital nets for finite
$p$. In the year 2001 Larcher and Pillichshammer identified a class of digital
nets for which the symmetrized version in the sense of Davenport has $L_2$
discrepancy of the order $\sqrt{\log N}/N$, which is best possible due to the
celebrated result of Roth. However, it remained open whether this discrepancy
bound also holds for the original digital nets without any modification.
  In the present paper we identify nets from the above mentioned class for
which the symmetrization is not necessary in order to achieve the optimal order
of $L_p$ discrepancy for all $p \in [1,\infty)$.
  Our findings are in the spirit of a paper by Bilyk from 2013, who considered
the $L_2$ discrepancy of lattices consisting of the elements $(k/N,\{k
\alpha\})$ for $k=0,1,\ldots,N-1$, and who gave Diophantine properties of
$\alpha$ which guarantee the optimal order of $L_2$ discrepancy.
