We present a systematic theory connecting the nonlocal response kernel of a homogeneous medium to its effective surface susceptibilities for an arbitrary curved interface. Starting from the most general tensorial nonlocal constitutive relation and combining a spatial moment expansion with a distributional thin-layer limit, we show that the full complexity of the interfacial response condenses, at leading order, into two surface susceptibilities ${\chi }_{\parallel }^s$ and ${\chi }_{\perp }^s$, given by the first normal moment of the bulk kernel over the exterior half-space. They carry the dimension of a length, coincide for a scalar kernel, and generically differ for a tensorial one, the anisotropy of the surface response being inherited from the longitudinal/transverse contrast of the bulk kernel. These quantities provide a constructive generalization of the Feibelman $d$-parameters to interfaces of arbitrary curvature; the next moment generates a dipole-layer term ${\partial }_n{\delta }_{\partial \Omega }$, and it is through the anisotropy of that layer that curvature first reaches the leading surface term, in a combination whose integral over any closed surface vanishes, as the parity of the layer mass requires; for a scalar kernel curvature is deferred one order further, the leading term then seeing the geometry only through the quadratic invariant $3H^2-4K$. Generalized Maxwell boundary conditions are established and compared with the classical Fresnel results, and the formalism is illustrated on the nonlocal two-dimensional Mie scattering of a dielectric cylinder.