In large spheroidal stellar systems, such as elliptical galaxies, one invariably finds a 106-109 M⊙ supermassive black hole at their centre. In contrast, within dwarf elliptical galaxies one predominantly observes a 105-107 M⊙ nuclear star cluster. To date, few galaxies have been found with both types of nuclei coexisting and even less have had the masses determined for both central components. Here, we identify one dozen galaxies housing nuclear star clusters and supermassive black holes whose masses have been measured. This doubles the known number of such hermaphrodite nuclei - which are expected to be fruitful sources of gravitational radiation. Over the host spheroid (stellar) mass range 108-1011 M⊙, we find that a galaxy's nucleus-to-spheroid (baryon) mass ratio is not a constant value but decreases from a few per cent to ∼0.3 per cent such that log[(MBH + MNC)/Msph] = -(0.39 ± 0.07) log[Msph/1010 M⊙] - (2.18 ± 0.07). Once dry merging commences and the nuclear star clusters disappear, this ratio is expected to become a constant value. As a byproduct of our investigation, we have found that the projected flux from resolved nuclear star clusters is well approximated with Sérsic functions having a range of indices from ∼0.5 to ∼3, the latter index describing the Milky Way's nuclear star cluster.