We introduce a classification scheme of the post-merger dynamics and gravitational-wave emission in binary neutron star mergers, after identifying a new mechanism by which a secondary peak in the gravitational-wave spectrum is produced. It is caused by a spiral deformation, the pattern of which rotates slower with respect to the double-core structure in the center of the remnant. This secondary peak is typically well separated in frequency from the secondary peak produced by a nonlinear interaction between a quadrupole and a quasi-radial oscillation. The new mechanism allows for an explanation of low-frequency modulations seen in a number of physical characteristics of the remnant, such as the central lapse function, the maximum density and the separation between the two cores, but also in the gravitational-wave amplitude. We find empirical relations for both types of secondary peaks between their gravitational-wave frequency and the compactness of nonrotating individual neutron stars, that exist for fixed total binary masses. These findings are derived for equalmass binaries without intrinsic neutron-star spin analyzing hydrodynamical simulations without magnetic field effects. Our classification scheme may form the basis for the construction of detailed gravitational-wave templates of the post-merger phase. We find that the quasi-radial oscillation frequency of the remnant decreases with the total binary mass. For a given merger event our classification scheme may allow to determine the proximity of the measured total binary mass to the threshold mass for prompt black-hole formation, which can, in turn, yield an estimate of the maximum neutron-star mass.