On the observational diagnostics to separate classical and disk-like bulges.

Flattened bulges with disk-like properties are considered to be the end product of secular evolution processes at work in the inner regions of galaxies. On the contrary, classical bulges are characterized by rounder shapes and thought to be similar to low-luminosity elliptical galaxies. We aim at testing the variety of observational diagnostics which are commonly adopted to separate classical from disk-like bulges in nearby galaxies. We select a sample of eight unbarred lenticular galaxies to be morphologically and kinematically undisturbed with no evidence of other components than bulge and disk. We analyze archival data of broad-band imaging from SDSS and integral-field spectroscopy from the ATLAS$^{\rm 3D}$ survey to derive the photometric and kinematic properties, line-strength indices, and intrinsic shape of the sample bulges. We argue that the bulge S\'ersic index is a poor diagnostics to discriminate different bulge types. We find that the combination of line-strength with either kinematic or photometric diagnostics does not provide a clear separation for half of the sample bulges. We include for the first time the intrinsic three-dimensional shape of bulges as a possible discriminant of their nature. All bulges turn out to be thick oblate spheroids, but only one has a flattening consistent with that expected for outer disks. We conclude that bulge classification may be difficult even adopting all observational diagnostics proposed so far and that classical and disk-like bulges could be more confidently identified by considering their intrinsic shape.