Observing the structure of quantized vortices can provide evidence for the pairing nature of a superfluid or superconductor and pinpoint its order parameter. Spin-triplet superfluid ${}^3$He supports a variety of vortices, calculated and identified so far in bulk fluid. We show numerically that the vortex core in ${}^3$He is strongly altered near a surface, resulting in a structure inhomogeneous along the vortex line. The effect is asymmetric with respect to the relative orientation of the core order parameter anisotropy axis and the surface normal. Coupling of spin and orbital degrees of freedom in the gradient energy of a triplet order parameter allows the surface to lift the degeneracy of two bulk vortex states connected by spontaneously broken inversion symmetry of the core. In a wide range of external conditions, the vortex structure at the surface is found to be completely different from that in bulk. As an implication, surface-limited vortex core observations in a triplet-candidate system may not reflect the bulk structure. We propose an experimental verification of the effect by measuring a transition in the vortex structure in thin slabs of superfluid ${}^3$He-B.