We present a discovery of an unusual unidirectionally rotating windmill scattering of electromagnetic waves by a magnetized gyromagnetic cylinder via an analytical theory for rigorous solution to fields and charges and an understanding of the underlying mathematical and physical mechanisms. Mathematically, the generation of nonzero off-diagonal components can break the symmetry of forward and backward scattering coefficients, producing unidirectional windmill scattering. Physically, this windmill scattering originates from the nonreciprocal unidirectional rotation of polarized magnetic charges on the surface of a magnetized gyromagnetic cylinder, which drives the scattering field to radiate outward in the radial direction and unidirectionally emit in the tangential direction. Interestingly, the unidirectional electromagnetic windmill scattering is insensitive to the excitation direction. Moreover, we also discuss the size dependence of unidirectional windmill scattering by calculating the scattering spectra of the gyromagnetic cylinder. These results are helpful for exploring and understanding novel interactions between electromagnetic waves and gyromagnetic materials or structures and offer deep insights for comprehending topological photonic states in gyromagnetic systems from the aspect of fundamental classical electrodynamics and electromagnetics.