We reveal unusually strong polarization sensitivity of electric and magnetic dipole resonances of high‐index dielectric nanoparticles placed on a metallic film. By employing dark‐field spectroscopy, we observe the polarization‐controlled transformation from high‐Q magnetic‐dipole scattering to broadband suppression of scattering associated with the electric dipole mode, and show numerically that it is accompanied by a strong enhancement of the respective fields by the nanoparticle. Our experimental data for silicon nanospheres are in an excellent agreement with both analytical calculations based on Green's function approach and the full‐wave numerical simulations. Our findings further substantiate dielectric nanoparticles as strong candidates for many applications in enhanced sensing, spectroscopy and nonlinear processes at the nanoscale.