We apply invertible field theories to study two questions in quantum field theory. Specifically, we study reflection-positive fully-extended invertible field theories on manifolds with twisted spin structures, which are computed as Anderson-dual bordism groups [1, 2]. In high energy physics, invertible field theories represent anomalies of quantum field theories. Our first application is toward ’t Hooft anomaly matching—a method first developed in the 1980s in which one treats anomalies as invariants of theories of interest and uses them to compute how quantum field theories change under physical processes. Specifically, we model three related processes around a form of spontaneous symmetry breaking via a charged order parameter using a twisted Gysin sequence of Anderson-dual bordism groups. We study the Smith maps of Madsen-Tillmann spectra that underlie the sequence, collecting examples and cataloging periodicities. Finally, we compute an extensive set of examples of physical interest and draw physical predictions from the results. In condensed matter physics, invertible field theories model the low energy field theories of symmetry-protected topological phases (SPTs). In this second application, we develop and compute homotopical free-to-interacting maps to compare two classifications of fermionic SPTs: those for free (i.e. non-interacting) models, and more general interacting classifications. These maps contribute to what has been a prolific line of research in the physics literature for the past fifteen years. Generalizing Freed--Hopkins [1], we construct maps from K-theory to twisted spin IFTs using T-duality and twisted versions of the spin orientation of K-theory [3]. We focus on two situations: weak phases [4, 5], which are SPTs protected by discrete translation symmetry, and primed phases [6], which are closely related to the famous tenfold way [7, 8], but which have a very different interacting classification. In the latter case, we demonstrate the dependence of the interacting classification on more than the Morita class of the symmetry algebra.