The Nysa-Polana dynamical family is one of the most complex groupings identified in the asteroid main belt. It has a generally twofold structure and is located in a region of the belt showing an unusual abundance of the fairly rare F-type taxonomic class. We present here new spectra of 22 family members, observed at ESO (European Southern Observatory) and at CASLEO (Complejo Astronomico El Leoncito, Argentina). We find evidence that the family is actually formed by two distinct groupings mutually overlapping in the space of orbital proper elements. The first group of objects, consisting of dark asteroids including several F-type members, can be named after its least-numbered member, 142 Polana. The second group consists of S-type asteroids and should be named after its most plausible least-numbered member, 878 Mildred. Some detailed physical analysis of the two groupings can be already attempted at this stage, including a reasonable reconstruction of the original ejection velocity field for the Mildred family. However, we face here a number of intriguing problems that deserve further analysis. For instance, it is not clear how to interpret the role of the two largest nominal members of the clan, 44 Nysa and 135 Hertha, as well as the presence of several large F-type asteroids apparently not related, but very close to the clan. Moreover, recent discoveries of common spectral features indicating hydrated minerals in the spectra of Nysa and Hertha add complexity to an already puzzling scenario.