In the last decade star clusters have been found in the centers of spiral galaxies across all Hubble types. We here present a spectroscopic study of the exceptionally bright (106-108 Lsolar) but compact (re~5 pc) nuclear star clusters in very late type spirals with the Ultraviolet and Visual Echelle Spectrograph at the VLT. We find that the velocity dispersions of the nine clusters in our sample range from 13 to 34 km s-1. Using photometric data from the Hubble Space Telescope WFPC2 and spherically symmetric dynamical models, we determine masses between 8×105 and 6×107 Msolar. The mass-to-light ratios range from 0.2 to 1.5 in the I band. This indicates a young mean age for most clusters, in agreement with previous studies. Given their high masses and small sizes, we find that nuclear clusters are among the objects with the highest mean surface density known (up to 105 Msolar pc-2). From their dynamical properties we infer that, rather than small bulges, the closest structural kin of nuclear clusters appear to be massive compact star clusters. This includes such different objects as globular clusters, ``super star clusters,'' ultracompact dwarf galaxies (UCDs), and the nuclei of dwarf elliptical galaxies. It is a challenge to explain why, despite the widely different current environments, all different types of massive star clusters share very similar and structural properties. A possible explanation links UCDs and massive globular clusters to nuclear star clusters through stripping of nucleated dwarf galaxies in a merger event. The extreme properties of this type of cluster would then be a consequence of the clusters' location in the centers of their respective host galaxies.