It is customary to construct the symmetric monoidal category \(\mathsf{Spt}\) of spectra from the symmetric monoidal category of \(\mathsf{Top}_\ast\) of pointed spaces by freely adjoining an inverse to \(S^1\) under smash product: \(\mathsf{Spt} = \mathsf{Top}_\ast[S^{-1}]\). Similarly, the category of genuine \(G\)-equivariant spectra for a finite group \(G\) and the category of motivic spectra over a smooth scheme \(S\) are constructed by inverting represesntation spheres, and the Tate sphere respectively. But this begs the question: how do we know which objects ought to be inverted? We suggest an alternative approach: rather than choosing certain objects to invert, one may formally adjoin duals to a certain class of objects which may be easier to identify. For example, one may adjoin "Spanier-Whitehead" duals to all finite CW-complexes on \(\mathsf{Top}_\ast\). The resulting category splits as a product of easily-defined factors, one factor of which is the usual category \(\mathsf{Spt}\) of spectra. Similar results apply in the equivariant and motivic settings, suggesting that duality properties may play an important role in determining, for a given "unstable homotopy theory", the "correct stable homotopy theory" associated to it.