  We have studied the complete set of dimension 5 and dimension 6 effective
operators involving the interaction of scalar, fermion and vector Dark Matter
(DM) with SM quarks and gluons, to explore the possibility to distinguish these
operators and characterise the spin of DM at the LHC. We have found that three
factors - the effective dimension of the operator, the structure of the SM part
of the operator and the parton densities of the SM particles connected to the
operator - uniquely define the shape of the (unobservable) invariant mass
distribution of the DM pair and, consequently, the shape of the (observable)
MET distribution related to it. Using $\chi^2$ analysis, we found that at the
LHC, with a luminosity of 300 fb$^{-1}$, certain classes of EFT operators can
be distinguished from each other. Hence, since DM spin is partly correlated
with the factors defining the shape of MET, the LHC can potentially shed a
light also on DM spin. We have also observed a drastic difference in the
efficiencies (up to two orders of magnitude) for large MET cuts scenarios with
different DM spin, thus indicating that the DM discovery potential strongly
depends on it. The study we perform here can be applied more generally than
within the EFT paradigm, where the DM mediator is not produced
on-the-mass-shell, such as the case of t-channel mediator or mediator with mass
below $2M_{DM}$, where the invariant mass of the DM pair is not fixed.
