We provide a theoretical analysis of some empirical facts about the second order spatiochromatic structure of natural images in color. In particular, we show that two simple assumptions on the covariance matrices of color images yield eigenvectors made by the Kronecker product of Fourier features times the triad given by luminance plus color opponent channels. The first of these assumptions is second order stationarity while the second one is commutativity between color correlation matrices. The validity of these assumptions and the predicted shape of the PCA components of color images are experimentally observed on two large image databases. As a by-product of this experimental study, we also provide novel data to support an exponential decay law of the spatiochromatic covariance between pairs of pixels as a function of their spatial distance.