The spheroid of the Sombrero galaxy, NGC 4594, is considered a prototype of classical, merger-built bulges. We use a Spitzer, IRAC 3.6 micron image to perform a detailed structural analysis of this galaxy. If one fits to this image only bulge and disc components, the bulge occupies a locus in the mass-size relation close to that of elliptical galaxies. When an outer stellar spheroid is added to improve the fit, the bulge Sersic index drops by a factor of ~ 2, and, if taken at face value, could mean that this bulge is actually a disc-like, pseudo-bulge, or a bar viewed end-on. The bulge effective radius and the bulge-to-total ratio also drop dramatically, putting the bulge in a position closer to that of bulges in the mass-size relation. We discuss implications from these findings, including the locus of the Sombrero bulge in the black hole mass vs. bulge mass relation. With this new bulge mass estimate, current dynamical estimates for the mass of the central black hole in Sombrero are more than 10 times larger than expected, if only the bulge mass is considered. A better agreement is found if the sum of bulge and outer spheroid masses is considered. Furthermore, residual images show the presence of a stellar ring and a stellar, inner ring or disc, with unprecedented clarity. We also show that Sombrero is an outlier in scaling relations of disc galaxies involving the disc, the spheroid and the globular cluster system, but not so when its structural components are considered independently. In this context, the globular cluster system of Sombrero might not be representative of disc galaxies. Finally, we discuss the possibility that Sombrero formed as an elliptical galaxy but accreted a massive disc, which itself has secularly evolved, resulting in a complex and peculiar system.