While navigating their environment, many animals keep track of their orientation over time. For example, in the Drosophila central complex, heading neurons track the fly’s orientation, similar to head direction cells in rodents. However, the circuit architecture that gives rise to these orientation tracking properties remains largely unknown in any species. I will describe a set of clockwise- and counterclockwise-shifting neurons in Drosophila whose wiring and calcium dynamics provide a means to rotate the heading neurons' signal clockwise or counterclockwise in the fly brain. I will further show that clockwise- and counterclockwise-shifting neurons each exist in two subtypes, with temporal and spatial profiles that suggest opposing roles for each subtype at the start and end of a turn. Shifting neurons are required for this heading system to properly track the fly's movements in the dark, and, when stimulated, are sufficient to shift the heading system in the expected direction. The central features of this circuit are analogous to models proposed for head-direction cells in rodents, and can in principle be extended to integrate other variables, including position. This orientation circuit may thus inform how neural systems, in general, perform addition