ABSTRACT We investigate the correlations between the black hole (BH) mass MBH, the velocity dispersion , the bulge mass MBu, the bulge average spherical density , and its spherical half-mass radius rh, constructing a database of 97 galaxies (31 core ellipticals, 17 power-law ellipticals, 30 classical bulges, and 19 pseudobulges) by joining 72 galaxies from the literature to 25 galaxies observed during our recent SINFONI BH survey. For the first time we discuss the full error covariance matrix. We analyze the well-known MBH- and MBH-MBu relations and establish the existence of statistically significant correlations between MBu and rh and anticorrelations between MBu and . We establish five significant bivariate correlations (MBH- - h, MBH- -rh, MBH-MBu- , MBH-MBu- h, MBH-MBu-rh) that predict MBH of 77 core and power-law ellipticals and classical bulges with measured and intrinsic scatter as small as dex and dex, respectively, or 0.26 dex when the subsample of 45 galaxies defined by Kormendy & Ho is considered. In contrast, pseudobulges have systematically lower MBH but approach the predictions of all of the above relations at spherical densities or scale lengths . These findings fit in a scenario of coevolution of BH and classical-bulge masses, where core ellipticals are the product of dry mergers of power-law bulges and power-law ellipticals and bulges the result of (early) gas-rich mergers and of disk galaxies. In contrast, the (secular) growth of BHs is decoupled from the growth of their pseudobulge hosts, except when (gas) densities are high enough to trigger the feedback mechanism responsible for the existence of the correlations between MBH and galaxy structural parameters.