  The magnetic properties of Ho-doped langasites (La:Ho)$_3$Ga$_5$SiO$_{14}$
are dominated by the Ising-like magnetic moments of the Ho$^{3+}$ ions. In
their saturated regime, the induced magnetic state breaks both time and space
inversion symmetries, leading to a novel linear magnetoelectric effect.
However, due to distortions induced by a shared Ga/Si occupancy of the 2d
sites, resolving the microscopic nature of the magnetic configuration remains a
difficult task. Here we combine polarized neutron diffraction and angular
dependent magnetization experiments to determine the local distortions of the
Ho$^{3+}$ magnetic moments in doped langasites
(La$_{1-x}$Ho$_x$)$_3$Ga$_5$SiO$_{14}$ with $x \approx 0.015$ and $x \approx
0.045$. We propose a model for a field-induced magnetic configuration with
arbitrary orientations of the local Ising axis of Ho$^{3+}$ in distorted
positions. The operations of broken local $C_2$ symmetry and rotations around
the trigonal $C_3$ axis connect different sites, restoring the global P321
symmetry of the crystal and simplifying the description of the magnetic
properties. The superposition of two distorted Ho$^{3+}$ positions connected by
$C_2$ symmetry determines the local magnetic susceptibility tensor, which no
longer appears Ising-like at low fields.
