We used several data sets from the Plateau de Bure Interferometer to map the dimethyl ether emission in Orion-KL with different arcsec spatial resolutions and different energy levels to compare with our previous methyl formate maps. Our data show remarkable similarity between the dimethyl ether (CH3OCH3) and the methyl formate (HCOOCH3) distributions even on a small scale (1.8"x0.8" or about 500 AU). This long suspected similarity, seen from both observational and theoretical arguments, is demonstrated with unprecedented confidence, with a correlation coefficient of maps of 0.8. A common precursor is the simplest explanation of our correlation. Comparisons with previous laboratory work and chemical models suggest the major role of grain surface chemistry and a recent release, probably with little processing, of mantle molecules by shocks. In this case the CH3O radical produced from methanol ice would be the common precursor (whereas ethanol, C2H5OH, is produced from the radical CH2OH). The alternative gas phase scheme, where protonated methanol CH3OH2+ is the common precursor to produce methyl formate and dimethyl ether through reactions with HCOOH and CH3OH, is also compatible with our data. Our observations cannot yet definitely allow a choice between the different chemical processes, but the tight correlation between the distributions of HCOOCH3 and CH3OCH3 strongly contrasts with the different behavior we observe for the distributions of ethanol and formic acid. This provides a very significant constraint on models.