Effect of rotationally invariant Hund's rule coupling on a magnetism of multiorbital Hubbard models is studied within a dynamical mean-field theory framework. Comparison of static magnetic susceptibilities and local densities of states of two- and three-orbital models of a complete rotationally invariant Coulomb interaction and a ``density-density{''} Hartree type interaction shows the different role of spin-flip interactions for different band fillings. In the particle-hole symmetric case, the Mott-Hubbard physics dominates due to the strong effective Coulomb interaction, while for the multiple electronic configurations away from half-filling (two electrons in the three-band model) the formation of local magnetic moments due to Hund's exchange interaction becomes the most significant effect for itinerant magnetic systems. A shift of the temperature of magnetic ordering due to the rotationally invariant Hund's rule coupling is found to be the largest in a three-orbital model with a two- electron occupancy where the single particle spectrum is metallic and is not sensitive to different forms of the Coulomb vertex. A larger enhancement of the effective mass in a model with a rotationally invariant interaction is discussed. In the half-filled case we find a drastic change in the density of states close to the Mott transition which is related to the spin-flip Kondo fluctuations in a degenerate orbital case, while the corresponding shift of the magnetic transition temperature is relatively small. It is shown that a change in the ground-state degeneracy due to a different symmetry of the Coulomb interaction in the density-density model leads to a breakdown of the quasiparticle peak at the Fermi level in the proximity of a Mott transition on the metallic side. We discuss the relevance of rotationally invariant Hund's interaction in the transition metal magnetism.