  We prove conjectures of Auroux stating that homological mirror symmetry for
very affine hypersurfaces respects certain natural symplectic operations (as
functors between partially wrapped Fukaya categories). These conjectures
concern compatibility between mirror symmetry for a very affine hypersurface
and its complement, itself also a very affine hypersurface. We find that the
complement of a very affine hypersurface has in fact two natural mirrors, one
of which is a derived scheme. These two mirrors are related via a non-geometric
equivalence mediated by Kn\"orrer periodicity; Auroux's conjectures require
some modification to take this into account. Our proof also uses new symplectic
techniques for gluing Liouville sectors which may be of independent interest.
