Understanding the nature of bulges in disc galaxies can provide important insights into the formation and evolution of galaxies. For instance, the presence of a classical bulge suggests a relatively violent history. In contrast, the presence of an inner disc instead (also referred to as a "pseudobulge") indicates the occurrence of secular evolution processes in the main disc. However, we still lack criteria to effectively categorise bulges, limiting our ability to study their impact on the evolution of the host galaxies. Here we present a recipe to separate inner discs from classical bulges by combining four different parameters from photometric and kinematic analyses: the bulge Sérsic index nb, the concentration index C20,50, the Kormendy (1977, ApJ, 217, 406) relation and the inner slope of the radial velocity dispersion profile ∇σ. With that recipe we provide a detailed bulge classification for a sample of 45 galaxies from the integral-field spectroscopic survey CALIFA. To aid in categorising bulges within these galaxies, we perform 2D image decomposition to determine bulge Sérsic index, bulge-to-total light ratio, surface brightness and effective radius of the bulge and use growth curve analysis to derive a new concentration index, C20,50. We further extract the stellar kinematics from CALIFA data cubes and analyse the radial velocity dispersion profile. The results of the different approaches are in good agreement and allow a safe classification for approximately 95% of the galaxies. In particular, we show that our new "inner" concentration index performs considerably better than the traditionally used C50,90 when yielding the nature of bulges. We also found that a combined use of this index and the Kormendy relation gives a very robust indication of the physical nature of the bulge.