Metal spinning is an incremental forming technique commonly employed in the production of hollow axisymmetric components. In recent years, asymmetric spinning processes have been developed to expand the range of component geometries achievable by the technique. However, most of these processes have employed a solid mandrel to set the target shape and provide internal support to the workpiece, thus requiring new tooling to be manufactured for every new part. Moreover, no studies have been performed on the link between the geometry of the target part and the achievable forming height. In this paper, the range of curvature in the target part's planform is used to quantify its degree of asymmetry, and the influence of this parameter on the formability of spun parts is investigated in a series of experimental trials. The hypothesis that the range of planform curvature predicts the likelihood of workpiece failure is tested. Methods to design the multi-pass toolpaths and the blanks required to spin both axisymmetric and asymmetric components without a mandrel are developed. The results show that increasing the degree of asymmetry of the target part only weakly influences the achievable forming height. This finding points to the potential of the technique to produce multiple geometries flexibly and to reduce the costs of prototyping and testing new sheet metal parts considerably.