For two-dimensional (2D) arrays of metallic nanorods arranged perpendicular to a substrate several methods have been proposed to determine the electromagnetic near-field distribution and the surface plasmon resonances, but an analytical approach to explain all optical features on the nanometer length scale has been missing to date. To fill this gap, we demonstrate here that the field distribution in such arrays can be understood on the basis of surface plasmon polaritons (SPPs) that propagate along the nanorods and form standing waves. Notably, SPPs couple laterally through their optical near fields, giving rise to collective surface plasmon (CSP) effects. Using the dispersion relation of such CSPs, we deduce the condition of standing-wave formation, which enables us to successfully predict several features, such as eigenmodes and resonances. As one such property and potential application, we show both theoretically and in an experiment that CSP propagation allows for polarization conversion and optical filtering in 2D nanorod arrays. Hence, these arrays are promising candidates for manipulating the light polarization on the nanometer length scale.