A head direction (HD) cell fires when an animal faces a particular azimuthal direction (Taube and Bassett 2003). Recurrent connections among HD cells create a ring-attractor network that continuously encodes HD (Ajabi et al. 2023). Indeed, in the Drosophila HD system, three rings interact to linearly integrate angular head velocity (AHV), with two of the rings receiving countervailing velocity-dependent inputs (Turner-Evans et al. 2017). In contrast, for the desert locust an alternative mechanism has been proposed (Zittrell et al. 2023; Pabst et al. 2024). There, velocity modulates synaptic strengths to integrate AHV. Here, we put forward a theoretical framework that distinguishes the two models of AHV integration by showing that AHV will differentially affect the activity of HD cells when the network operates via synaptic modulation or via additive inputs. This framework allows us to infer the ring-attractor structure from single-cell activity without reference to the cells' structural connectivity. Applied to Drosophila, our framework deduces the presence of three interacting rings, in agreement with the findings of Turner-Evans et al. (2017). In zebrafish, an activity bump rotates across anatomical structures as the HD changes (Petrucco et al. 2023), but the exact structure of the ring-attractor is still unknown. When we apply our approach to the zebrafish data, we find that the zebrafish HD system is composed of three functional rings, too, even though these rings are only partially segregated anatomically. Our findings suggest that HD systems exhibit a high similarity across distant species and show that our theoretical approach holds promise for inferring the ring-attractor structures in other species, too.