We have measured the resonance strengths and energies for dielectronic recombination (DR) of Fe XX forming Fe XIX via N=2 --> N'=2 (Delta_N=0) core excitations. We have also calculated the DR resonance strengths and energies using AUTOSTRUCTURE, HULLAC, MCDF, and R-matrix methods, four different state-of-the-art theoretical techniques. On average the theoretical resonance strengths agree to within <~10% with experiment. However, the 1 sigma standard deviation for the ratios of the theoretical-to-experimental resonance strengths is >~30% which is significantly larger than the estimated relative experimental uncertainty of <~10%. This suggests that similar errors exist in the calculated level populations and line emission spectrum of the recombined ion. We confirm that theoretical methods based on inverse-photoionization calculations (e.g., undamped R-matrix methods) will severely overestimate the strength of the DR process unless they include the effects of radiation damping. We also find that the coupling between the DR and radiative recombination (RR) channels is small. We have used our experimental and theoretical results to produce Maxwellian-averaged rate coefficients for Delta_N=0 DR of Fe XX. For kT>~1 eV, which includes the predicted formation temperatures for Fe XX in an optically thin, low-density photoionized plasma with cosmic abundances, our experimental and theoretical results are in good agreement. We have also used our R-matrix results, topped off using AUTOSTRUCTURE for RR into J>=25 levels, to calculate the rate coefficient for RR of Fe XX. Our RR results are in good agreement with previously published calculations.