We report on the thorough investigation of light emitting diodes (LEDs) made of core-shell nanorods (NRs) with InGaN/GaN quantum wells (QWs) in the outer shell, which are grown on patterned substrates by metal-organic vapor phase epitaxy. The multi-bands emission of the LEDs covers nearly the whole visible region, including UV, blue, green, and orange ranges. The intensity of each emission is strongly dependent on the current density, however the LEDs demonstrate a rather low color saturation. Based on transmission electron microscopy data and comparing them with electroluminescence and photoluminescence spectra measured at different excitation powers and temperatures, we could identify the spatial origination of each of the emission bands. We show that their wavelengths and intensities are governed by different thicknesses of the QWs grown on different crystal facets of the NRs as well as corresponding polarization-induced electric fields. Also the InGaN incorporation strongly varies along the NRs, increasing at their tips and corners, which provides the red shift of emission. With increasing the current, the different QW regions are activated successively from the NR tips to the side-walls, resulting in different LED colors. Our findings can be used as a guideline to design effectively emitting multi-color NR-LEDs.