  We recapture Douglas' framework for twisted parametrized stable homotopy
theory in the language of $\infty$- categories. A twisted spectrum is
essentially a section of a bundle of presentable stable $\infty$-categories
whose fiber is the $\infty$-category of spectra, a perspective we refine in
this work. We recover some of Douglas' results on classifications of such
bundles, as well as further make precise the identification of categories of
twisted spectra with module categories over Thom spectra in the pointed
connected case. Furthermore, we examine subtle aspects of the functoriality
which arise by virtue of being fibered over the Brauer space of the sphere
spectrum. Extending beyond the scope of Douglas's work, we introduce a total
category of twisted spectra over a fixed space, where we allow the twists to
vary, and show that these total categories themselves satisfy the essential
features of a 6-functor formalism. We also introduce an $(\infty, 2)$-category
of twisted spectra, and use this framework to discuss duality theory for these
type of objects.
