  We study twisted derived equivalences for schemes in the setting of spectral
algebraic geometry. To this end, we introduce the notion of a twisted
equivalence and show that a twisted equivalence for perfect spectral algebraic
stacks admitting a quasi-finite presentation supplies an equivalence between
the stacks, which compensate for the failure of twisted derived equivalences
for non-affine schemes to provide an isomorphism of the schemes. In the case of
(not necessarily connective) commutative ring spectra, we also prove a spectral
analogue of Rickard's theorem, which shows that a derived equivalence of
associative rings induces an isomorphism between their centers.
