  In the zero-field-cooled exchange bias (ZEB) effect the unidirectional
magnetic anisotropy is set at low temperatures even when the system is cooled
in the absence of external magnetic field. La$_{1.5}$Sr$_{0.5}$CoMnO$_{6}$
stands out as presenting the largest ZEB reported so far, while for
La$_{1.5}$Ca$_{0.5}$CoMnO$_{6}$ the exchange bias field ($H_{EB}$) is one order
of magnitude smaller. Here we show that La$_{1.5}$Ba$_{0.5}$CoMnO$_{6}$ also
exhibits a pronounced shift of its magnetic hysteresis loop, with intermediate
$H_{EB}$ value in respect to Ca- and Sr-doped samples. In order to figure out
the microscopic mechanisms responsible for this phenomena, these compounds were
investigated by means of synchrotron X-ray powder diffraction, Raman
spectroscopy, muon spin rotation and relaxation, AC and DC magnetization, X-ray
absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD). The
parent compound La$_{2}$CoMnO$_{6}$ was also studied for comparison, as a
reference of a non-ZEB material. Our results show that the Ba-, Ca- and
Sr-doped samples present a small amount of phase segregation, and that the ZEB
effect is strongly correlated to the system's structure. We also observed that
mixed valence states Co$^{2+}$/Co$^{3+}$ and Mn$^{4+}$/Mn$^{3+}$ are already
present at the La$_{2}$CoMnO$_{6}$ parent compound, and that
Ba$^{2+}$/Ca$^{2+}$/Sr$^{2+}$ partial substitution at La$^{3+}$ site leads to a
large increase of Co average valence, with a subtle augmentation of Mn formal
valence. Estimates of the Co and Mn valences from the $L$-edge XAS indicate the
presence of oxygen vacancies in all samples (0.05$\leq \delta \leq$0.1). Our
XMCD results show a great decrease of Co moment for the doped compounds, and
indicate that the shift of the hysteresis curves for these samples is related
to uncompensated antiferromagnetic coupling between Co and Mn.
