  We construct a new kinematical variable that is able to fully reconstruct the
absolute value, and partially reconstruct the sign, of the angular distribution
in the center of momentum system of a decaying particle in certain cases where
the center of momentum system is only known up to a two-fold ambiguity. After
making contact with Drell-Yan production at the Large Hadron Collider, we apply
this method to the pair-production of dark matter in association with two
charged leptons at the International Linear Collider and show that for a small
intermediate width, perfect agreement is found with the true angular
distribution in the absence of initial state radiation. In the presence of
initial state radiation, we find that the modification to the angular
distributions is small for most angles and that the different spin combinations
should still be distinguishable. This enables us to determine the spin of the
mother particle and the dark matter particle in certain cases.
