Recent experimental and theoretical studies have highlighted the possible role of an electronic nematic liquid in underdoped cuprate superconductors. We calculate, within a model of d-wave superconductor with Hubbard correlations, the spin susceptibility in the case of a small explicitly broken rotational symmetry of the underlying lattice. We then exhibit how the induced spin response asymmetry is strongly enhanced by correlations as one approaches the instability to stripe order. In the disorder-induced stripe phase, impurities become spin nematogens with a C2 symmetric impurity resonance state, and the disorder-averaged spin susceptibility remains only C2 symmetric at low energies, similar to recent data from neutron scattering experiments on underdoped YBCO.