We focus on the infinite-temperature dynamical spin structure factor of the asymmetric Hatsugai-Kohmoto model, a relative of the asymmetric Hubbard model. It is characterized by distinct single-particle energies for the two spin species, which interact with each other through a contact interaction in momentum space. We evaluate its spin structure factor exactly and follow the evolution of its excitation spectrum for all fillings and interactions, identifying signatures of the Mott transition and fingerprints of the asymmetric hoppings. The longitudinal spin structure factor exhibits soundlike and interaction-induced gapped excitations, whose number gets doubled in the presence of hopping asymmetry. The transverse response displays the competition of interaction- and asymmetry-induced gaps and results in a quadratic excitation branch at their transition. The complete asymmetric case features a momentum-independent dynamical structure factor, characteristic of transitions involving a flat band.