  The notion of symmetrization, also known as Davenport's reflection principle,
is well known in the area of the discrepancy theory and quasi-Monte Carlo (QMC)
integration. In this paper we consider applying a symmetrization technique to a
certain class of QMC point sets called digital nets over $\mathbb{Z}_{b}$.
Although symmetrization has been recognized as a geometric technique in the
multi-dimensional unit cube, we give another look at symmetrization as a
geometric technique in a compact totally disconnected abelian group with dyadic
arithmetic operations. Based on this observation we generalize the notion of
symmetrization from base 2 to an arbitrary base $b\in \mathbb{N}$, $b\ge 2$.
Subsequently, we study the QMC integration error of symmetrized digital nets
over $\mathbb{Z}_{b}$ in a reproducing kernel Hilbert space. The result can be
applied to component-by-component construction or Korobov construction for
finding good symmetrized (higher order) polynomial lattice rules which achieve
high order convergence of the integration error for smooth integrands at the
expense of an exponential growth of the number of points with the dimension.
Moreover, we consider two-dimensional symmetrized Hammersley point sets in
prime base $b$, and prove that the minimum Dick weight is large enough to
achieve the best possible order of $L_p$ discrepancy for all $1\le p< \infty$.
