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We present the rest-frame optical and infrared colours of a complete sample of galaxies from the Spitzer Wide-Area Infrared Extragalactic (SWIRE) Legacy Survey and the Sloan Digital Sky Survey (SDSS). We discuss the optical and infrared colours of our sample and analyse in detail the contribution of dusty star-forming galaxies and active galactic nuclei (AGN) to optically selected red sequence galaxies. We propose that the optical colour and infrared colour of galaxies in our sample are determined primarily by a bulge-to-disc ratio. The colour is found to be sensitive to the bulge-to-disc ratio for disc-dominated galaxies, whereas the colour is more sensitive for bulge-dominated systems. We identify ∼18 per cent (195 sources) of our sample as having red optical colours and infrared excess. Typically, the infrared luminosities of these galaxies are found to be at the high end of star-forming galaxies with blue optical colours. Using emission-line diagnostic diagrams, 78 are found to have an AGN contribution and 117 are identified as star-forming systems. The red colour of the star-forming galaxies could be explained by extinction. However, their high optical luminosities cannot. We conclude that they have a significant bulge component. The number densities of optically red star-forming galaxies are found to correspond to ∼13 per cent of the total number density of our sample. In addition, these systems contribute ∼13 per cent of the total optical luminosity density, and 28 per cent of the total infrared luminosity density of our SWIRE/SDSS sample. These objects may reduce the need for ‘dry mergers’. [ABSTRACT FROM AUTHOR]