  It has been suggested that several small-scale structure anomalies in
$\Lambda$CDM cosmology can be solved by strong self-interaction between dark
matter particles. It was shown by Braaten and Hammer that the presence of a
near threshold S-wave resonance can make the scattering cross section at
nonrelativistic speeds come close to saturating the unitarity bound. This can
result in the formation of a stable bound state of two asymmetric dark matter
particles (we call the bound state as darkonium). Laha and Braaten studied the
nuclear recoil energy spectrum in dark matter direct detection experiments due
to this incident bound state. Here we study the angular recoil spectrum, and
show that it is uniquely determined up to normalization by the S-wave
scattering length. Observing this angular recoil spectrum in a dark matter
directional detection experiment will uniquely determine many of the low-energy
properties of dark matter independent of the underlying dark matter
microphysics.
