  The search of the optimal constant for a generalized Wirtinger inequality in
an interval consists in minimizing the $p$-norm of the derivative among all
functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero
average. Symmetry properties of minimizers have attracted great attention in
mathematical literature in the last decades, leading to a precise
characterization of symmetry and asymmetry regions.
  In this paper we provide a proof of the symmetry result without computer
assisted steps, and a proof of the asymmetry result which works as well for
local minimizers. As a consequence, we have now a full elementary description
of symmetry and asymmetry cases, both for global and for local minima.
  Proofs rely on appropriate nonlinear variable changes.
