  We develop the concept of twisted ambidexterity in a parametrized presentably
symmetric monoidal $\infty$-category, which generalizes the notion of
ambidexterity by Hopkins and Lurie and the Wirthm\"uller isomorphisms in
equivariant stable homotopy theory, and is closely related to Costenoble-Waner
duality. Our main result establishes the parametrized $\infty$-category of
genuine $G$-spectra for a compact Lie group $G$ as the universal example of a
presentably symmetric monoidal $\infty$-category parametrized over $G$-spaces
which is both stable and satisfies twisted ambidexterity for compact
$G$-spaces. We further extend this result to the settings of orbispectra and
proper genuine $G$-spectra for a Lie group $G$ which is not necessarily
compact.
