  We reveal how to exploit joint spatial-electromagnetic duality symmetries to
obtain invariant scattering properties (including extinction, scattering,
absorption) of self-dual scattering systems for incident waves of arbitrary
polarizations. The electromagnetic duality ensures the helicity preservation
along all scattering directions, and thus intrinsically eliminates the
interferences between the two scattering channels originating from the
circularly polarized components of incident waves. This absence of interference
directly secures invariant scattering properties for all polarizations located
on the same latitude circle of the Poincar\'{e} sphere, which are characterized
by polarization ellipses of the same eccentricity and handedness. Further
incorporations of mirror and/or inversion symmetries would lead to such
invariance throughout the whole Poincar\'{e} sphere, guaranteeing invariant
scattering properties for all polarizations. Simultaneous exploitations of
composite symmetries of different natures render an extra dimension of freedom
for scattering manipulations, offering new insights for both fundamental
explorations and optical device engineering related to symmetry dictated
light-matter interactions.
