  Double perovskite iridates $A_2$ZnIrO$_6$ ($A$ = alkaline or lanthanide) show
complex magnetic behaviors ranging from weak ferromagnetism to successive
antiferromagnetic transitions. Here we report the static ($dc$) and dynamic
($ac$) magnetic susceptibility, and neutron powder diffraction measurements for
$A$ = La and Nd compounds to elucidate the magnetic ground state. Below 10~K,
the $A$ = La compound is best described as canted iridium moments in an
antiferromagnet arrangement with a propagation vector \textbf{k} = 0 and a net
ferromagnetic component along the $c$-axis. On the other hand, Nd$_2$ZnIrO$_6$
is described well as an antiferromagnet with a propagation vector \textbf{k} =
(1/2~1/2~0) below $T_\mathrm{N} \sim$ 17 K. Scattering from both the Nd and Ir
magnetic sublattices were required to describe the data and both were found to
lie almost completely within the $ab$-plane. $Dc$ susceptibility revealed a
bifurcation between the zero-field-cooled and field-cooled curves below
$\sim$13 K in Nd$_2$ZnIrO$_6$. A glassy state was ruled out by $ac$
susceptibility but detailed magnetic isotherms revealed the opening of the loop
below 13~K. These results suggest a delicate balance exists between the
Dzyaloshinskii-Moriya, crystal field schemes, and $d$-$f$ interaction in this
series of compounds.
