Context. The intrinsic shape of galactic bulges in nearby galaxies provides crucial information to separate bulge types. Aims: We aim to derive accurate constraints to the intrinsic shape of bulges to provide new clues on their formation mechanisms and set new limitations for future simulations. Methods: We retrieved the intrinsic shape of a sample of CALIFA bulges using a statistical approach. Taking advantage of GalMer numerical simulations of binary mergers we estimated the reliability of the procedure. Analyzing the i-band mock images of resulting lenticular remnants, we studied the intrinsic shape of their bulges at different galaxy inclinations. Finally, we introduced a new (B/A, C/A) diagram to analyze possible correlations between the intrinsic shape and the properties of bulges. Results: We tested the method on simulated lenticular remnants, finding that for galaxies with inclinations of 25° ≤ θ ≤ 65° we can safely derive the intrinsic shape of their bulges. We found that our CALIFA bulges tend to be nearly oblate systems (66%), with a smaller fraction of prolate spheroids (19%), and triaxial ellipsoids (15%). The majority of triaxial bulges are in barred galaxies (75%). Moreover, we found that bulges with low Sérsic indices or in galaxies with low bulge-to-total luminosity ratios form a heterogeneous class of objects; additionally, bulges in late-type galaxies or in less massive galaxies have no preference for being oblate, prolate, or triaxial. On the contrary, bulges with high Sérsic index, in early-type galaxies, or in more massive galaxies are mostly oblate systems. Conclusions: We concluded that various evolutionary pathways may coexist in galaxies, with merging events and dissipative collapse being the main mechanisms driving the formation of the most massive oblate bulges and bar evolution reshaping the less massive triaxial bulges.