Asteroids can be classified into several groups based on their spectral reflectance. Among these groups, the one belonging to the L-class in the taxonomic classification based on visible and near-infrared spectra exhibit several peculiar properties. First, their near-infrared spectrum is characterized by a strong absorption band interpreted as the diagnostic of a high content of the FeO bearing spinel mineral. This mineral is one of the main constituents of Calcium-Aluminum-rich Inclusions (CAI) the oldest mineral compounds found in the solar system. In polarimetry, they possess an uncommonly large value of the inversion angle incompatible with all known asteroid belonging to other taxonomical classes. Asteroids found to possess such a high inversion angle are commonly called Barbarians based on the first asteroid on which this property was first identified, (234) Barbara. In this paper we present the results of an extensive campaign of polarimetric and spectroscopic observations of L-class objects. We have derived phase-polarization curves for a sample of 7 Barbarians, finding a variety of inversion angles ranging between 25 and 30°. Spectral reflectance data exhibit variations in terms of spectral slope and absorption features in the near-infrared. We analyzed these data using a Hapke model to obtain some inferences about the relative abundance of CAI and other mineral compounds. By combining spectroscopic and polarimetric results, we find evidence that the polarimetric inversion angle is directly correlated with the presence of CAI, and the peculiar polarimetric properties of Barbarians are primarily a consequence of their anomalous composition.