The issue of reversibility in hydromechanical sprinklers that auto-rotate while ejecting fluid from S-shaped tubes raises fundamental questions that remain unresolved. Here, we report on precision experiments that reveal robust and persistent reverse rotation under suction and a model that accounts for the observed motions. We implement an ultralow friction bearing in an apparatus that allows for free rotation under ejection and suction for a range of flow rates and arbitrarily long times. Flow measurements reveal a rocketlike mechanism shared by the reverse and forward modes that involves angular momentum flux, whose subtle manifestation in the reverse case stems from centrifugal effects for flows in curved conduits. These findings answer Feynman's long-standing question by providing quantitatively accurate explanations of both modes, and they suggest further inquiries into flux-based force generation and the roles of geometry and Reynolds number.