  A key parameter for the low-temperature magnetic coupling of in dinuclear
lanthanide single-molecule magnets (SMMs) is the barrier $U_{FA}$ resulting
from the exchange and dipole interactions between the two $4f$ moments. Here we
extend the pseudospin model previously used to describe the ground state of
dinuclear endofullerenes to account for variations in the orientation of the
single-ion anisotropy axes and apply it to the two SMMs Dy$_2$ScN@C$_{80}$ and
Dy$_2$TiC@C$_{80}$. While x-ray magnetic circular dichroism (XMCD) indicates
the same $J_z=15/2$ Dy groundstate in both molecules, the Dy-Dy coupling
strength and the stability of magnetization is distinct. We demonstrate that
both the magnitude of the barrier $U_{FA}$ and the angle between the two $4f$
moments are determined directly from precise temperature-dependent
magnetization data to an accuracy better than $1^{\circ}$. The experimentally
found angles between the $4f$ moments are in excellent agreement with
calculated angles between the quantisation axes of the two Dy ions. Theory
indicates a larger deviation of the orientation of the Dy magnetic moments from
the Dy bond axes to the central ion in Dy$_2$TiC@C$_{80}$. This may explain the
lower stability of the magnetisation in Dy$_2$TiC@C$_{80}$, although it
exhibits a $\sim 49\%$ stronger exchange coupling than in Dy$_2$ScN@C$_{80}$.
