We study geometric modular flows in two-dimensional conformal field theories. We explore which states exhibit a geometric modular flow with respect to a causally complete subregion and, conversely, how to construct a state from a given geometric modular flow. Given suitable boundary conditions, we find that generic geometric modular flows in the Rindler wedge are conformally equivalent. Based on this insight, we show how conformal unitaries can be used to explicitly construct a state for each flow. We analyze these states, deriving general formulas for the energy density and entanglement entropy. We also consider geometric flows beyond the Rindler wedge setting, and in higher dimensions.