  We consider three different questions related to the Steklov and mixed
Steklov problems on surfaces. These questions are connected by the techniques
that we use to study them, which exploit symmetry in various ways even though
the surfaces we study do not necessarily have inherent symmetry.
  In the spirit of the celebrated Hersch-Payne-Schiffer and Weinstock
inequalities for Steklov eigenvalues, we obtain a sharp isoperimetric
inequality for the mixed Steklov eigenvalues considering the interplay between
the eigenvalues of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalues.
  In 1980, Bandle showed that the unit disk maximizes the $k$th nonzero
normalized Steklov eigenvalue on simply connected domains with rotational
symmetry of order $p$ when $k\le p-1$. We discuss whether the disk remains the
maximizer in the class of simply connected rotationally symmetric domains when
$k\geq p$. In particular, we show that for $k$ large enough, the upper bound
converges to the Hersch-Payne-Schiffer upper bound.
  We give full asymptotics for mixed Steklov problems on arbitrary surfaces,
assuming some conditions at the meeting points of the Steklov boundary with the
Dirichlet or Neumann boundary.
