We show that there is a new class of gas tails-slingshot tails-that form as a subhalo (i.e., a subcluster or early-type cluster galaxy) moves away from the cluster center toward the apocenter of its orbit. These tails can point perpendicular or even opposite to the subhalo direction of motion, not tracing the recent orbital path. Thus, the observed tail direction can be misleading, and we caution against naive conclusions regarding the subhalo's direction of motion based on the tail direction. A head-tail morphology of a galaxy's or subcluster's gaseous atmosphere is usually attributed to ram pressure stripping, and the widely applied conclusion is that gas stripped tail traces the most recent orbit. However, during the slingshot tail stage, the subhalo is not being ram pressure stripped (RPS) and the tail is shaped by tidal forces more than just the ram pressure. Thus, applying a classic RPS scenario to a slingshot tail leads not only to an incorrect conclusion regarding the direction of motion but also to incorrect conclusions regarding the subhalo velocity, expected locations of shear flows, instabilities, and mixing. We describe the genesis and morphology of slingshot tails using data from binary cluster merger simulations and discuss their observable features and how to distinguish them from classic RPS tails. We identify three examples from the literature that are not RPS tails but slingshot tails and discuss other potential candidates.