  We determine the geometry of the target spaces of supersymmetric
non-relativistic particles with torsion and magnetic couplings, and with
symmetries generated by the fundamental forms of G-structures for $G= U(n),
SU(n), Sp(n), Sp(n)\cdot Sp(1), G_2$ and $Spin(7)$. We find that the
Killing-Yano equation, which arises as a condition for the invariance of the
worldline action, does not always determine the torsion coupling uniquely in
terms of the metric and fundamental forms. We show that there are several
connections with skew-symmetric torsion for $G=U(n), SU(n)$ and $G_2$ that
solve the invariance conditions. We describe all these compatible connections
for each of the $G$-structures and explain the geometric nature of the
couplings.
