  The Fe electronic structure and magnetism in (i) monoclinic Ca$_2$FeReO$_6$
with a metal-insulator transition at $T_{MI} \sim 140$ K and (ii) quasi-cubic
half-metallic Ba$_2$FeReO$_6$ ceramic double perovskites are probed by soft
x-ray absorption spectroscopy (XAS) and magnetic circular dichroism (XMCD).
These materials show distinct Fe $L_{2,3}$ XAS and XMCD spectra, which are
primarily associated with their different average Fe oxidation states (close to
Fe$^{3+}$ for Ca$_2$FeReO$_6$ and intermediate between Fe$^{2+}$ and Fe$^{3+}$
for Ba$_2$FeReO$_6$) despite being related by an isoelectronic
(Ca$^{2+}$/Ba$^{2+}$) substitution. For Ca$_2$FeReO$_6$, the powder-averaged Fe
spin moment along the field direction ($B = 5$ T), as probed by the XMCD
experiment, is strongly reduced in comparison with the spontaneous Fe moment
previously obtained by neutron diffraction, consistent with a scenario where
the magnetic moments are constrained to remain within an easy plane. For $B=1$
T, the unsaturated XMCD signal is reduced below $T_{MI}$ consistent with a
magnetic transition to an easy-axis state that further reduces the
powder-averaged magnetization in the field direction. For Ba$_2$FeReO$_6$, the
field-aligned Fe spins are larger than for Ca$_2$FeReO$_6$ ($B=5$ T) and the
temperature dependence of the Fe magnetic moment is consistent with the
magnetic ordering transition at $T_C^{Ba} = 305$ K. Our results illustrate the
dramatic influence of the specific spin-orbital configuration of Re $5d$
electrons on the Fe $3d$ local magnetism of these Fe/Re double perovskites.
