At the (001) surface of the alloy Ni₅₀Mn₅₀, a noncollinear spin density is observed in real space by spin-polarized scanning tunneling microscopy. The spin density of individual atoms also varies in both size and direction as a function of bias voltage, indicating a noncollinearity in the energy domain. The noncollinearity is driven by a surface reconstruction which breaks the otherwise high surface symmetry. First-principles electronic-structure calculations support the experimental observations and evidence the interplay of reconstruction and spin-orbit coupling.