  The analysis of a physical problem simplifies considerably when one uses a
suitable coordinate system. We apply this approach to the discrete-time quantum
walks with coins given by $2j+1$-dimensional Wigner rotation matrices (Wigner
walks), a model which was introduced in T. Miyazaki et al., Phys. Rev. A 76,
012332 (2007). First, we show that from the three parameters of the coin
operator only one is physically relevant for the limit density of the Wigner
walk. Next, we construct a suitable basis of the coin space in which the limit
density of the Wigner walk acquires a much simpler form. This allows us to
identify various dynamical regimes which are otherwise hidden in the standard
basis description. As an example, we show that it is possible to find an
initial state which reduces the number of peaks in the probability distribution
from generic $2j+1$ to a single one. Moreover, the models with integer $j$ lead
to the trapping effect. The derived formula for the trapping probability
reveals that it can be highly asymmetric and it deviates from purely
exponential decay. Explicit results are given up to the dimension five.
