Simulations of self-interacting dark matter predict that dark matter should lag behind galaxies during a collision. If the interaction is mediated by a high-mass force carrier, the distribution of dark matter can also develop asymmetric dark matter tails. To search for this asymmetry, we compute the gravitational lensing properties of a mass distribution with a free skewness parameter. We apply this to the dark matter around the four central galaxies in cluster Abell 3827. In the galaxy whose dark matter peak has previously been found to be offset, we tentatively measure a skewness s = 0.23(-0.22)(+0.05) in the same direction as the peak offset. Our method may be useful in future gravitational lensing analyses of colliding galaxy clusters and merging galaxies.