ABSTRACT This paper studies pseudo-bulges (P-bulges) and classical bulges (C-bulges) in Sloan Digital Sky Survey (SDSS) central galaxies using the new bulge indicator ΔΣ1, which measures relative central stellar-mass surface density within 1 kpc. We compare ΔΣ1 to the established bulge-type indicator Δ〈μe〉 from Gadotti (2009) and show that classifying by ΔΣ1 agrees well with Δ〈μe〉. ΔΣ1 requires no bulge–disc decomposition and can be measured on SDSS images out to z = 0.07. Bulge types using it are mapped on to 20 different structural and stellar-population properties for 12 000 SDSS central galaxies with masses 10.0 < log M*/M⊙ < 10.4. New trends emerge from this large sample. Structural parameters show fairly linear log–log relations versus ΔΣ1 and Δ〈μe〉 with only moderate scatter, while stellar-population parameters show a highly non-linear ‘elbow’ in which specific star formation rate remains roughly flat with increasing central density and then falls rapidly at the elbow, where galaxies begin to quench. P-bulges occupy the low-density end of the horizontal arm of the elbow and are universally star forming, while C-bulges occupy the elbow and the vertical branch and exhibit a wide range of star formation rates at a fixed density. The non-linear relation between central density and star formation rate has been seen before, but this mapping on to bulge class is new. The wide range of star formation rates in C-bulges helps to explain why bulge classifications using different parameters have sometimes disagreed in the past. The elbow-shaped relation between density and stellar indices suggests that central structure and stellar populations evolve at different rates as galaxies begin to quench.