  We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb
C)$ of $n\times n$ matrices with real eigenvalues interpolating the complex
conjugation and the transpose. We deduce from it a stratified homeomorphism
between the space of $n\times n$ real matrices with real eigenvalues and the
space of $n\times n$ symmetric matrices with real eigenvalues, which restricts
to a real analytic isomorphism between individual $\mathrm{GL}_n(\mathbb
R)$-adjoint orbits and $\mathrm{O}_n(\mathbb C)$-adjoint orbits. We also
establish similar results in more general settings of Lie algebras of classical
types and quiver varieties. To this end, we prove a general result about
involutions on hyper-K\"ahler quotients of linear spaces. We provide
applications to the (generalized) Kostant-Sekiguchi correspondence,
singularities of real and symmetric adjoint orbit closures, and Springer theory
for real groups and symmetric spaces.
