The interplay between mathematics and physics mutually affects both disciplines, and is the subject of ongoing research. Here we study the interplay in the empirical context of an MSc program for practicing high-school physics teachers putting strong emphasis on advanced physics courses taught by active physicists, together with pedagogical content courses taught by physics education researchers. Which aspects of the interplay between mathematics and physics do teacher instructors wish to convey to the teachers? We held semi-structured interviews with the physics instructors of the program, focusing on their goals for their courses. Thematic analysis of the interviews exposed several aspects of the interplay that instructors reported on integrating in their teaching, resulting in a characterization consisting of three families. The first strengthens established aspects in the literature: coherence, a tool for calculation, and mathematical derivation of new physics results (extraction). The second highlights new facets of existing aspects: the universality of geometry in physics, limits of approximations as invitations to explore, and simulations as virtual experiments (representations). The third comprises aspects that are less prominent in the physics education literature: concretization of abstract mathematics in physics contexts, and conflict as a mechanism for discovery (e.g., the Lorentz force’s dependence on velocity conflicting with Galilean invariance). Our findings provide a framework bridging tertiary physics and teaching across all levels—aspects of the interplay between mathematics and physics that are manifest in advanced physics could inform curricular design for high-school and beyond, and for teacher education.