  We show that over an algebraically closed field of characteristic not equal
to 2, homological projective duality for smooth quadric hypersurfaces and for
double covers of projective spaces branched over smooth quadric hypersurfaces
is a combination of two operations: one interchanges a quadric hypersurface
with its classical projective dual and the other interchanges a quadric
hypersurface with the double cover branched along it.
