  We clarify the relationship between Grothendieck duality \`a la Neeman and
the Wirthm\"uller isomorphism \`a la Fausk-Hu-May. We exhibit an interesting
pattern of symmetry in the existence of adjoint functors between compactly
generated tensor-triangulated categories, which leads to a surprising
trichotomy: There exist either exactly three adjoints, exactly five, or
infinitely many. We highlight the importance of so-called relative dualizing
objects and explain how they give rise to dualities on canonical subcategories.
This yields a duality theory rich enough to capture the main features of
Grothendieck duality in algebraic geometry, of generalized Pontryagin-Matlis
duality \`a la Dwyer-Greenlees-Iyengar in the theory of ring spectra, and of
Brown-Comenetz duality \`a la Neeman in stable homotopy theory.
