Reported here are transverse single-spin asymmetries (AN) in the production of charged hadrons as a function of transverse momentum (pT) and Feynman-x (xF) in polarized p↑+p, p↑+Al, and p↑+Au collisions at √sNN=200  GeV. The measurements have been performed at forward and backward rapidity (1.4<|η|<2.4) over the range of 1.5  GeV/c0) in p↑+p collisions, whereas the p↑+Al and p↑+Au results show smaller asymmetries. This finding provides new opportunities to investigate the origin of transverse single-spin asymmetries and a tool to study nuclear effects in p+A collisions.