Indifference to symmetry in Hrushovski's ab initio construction

Denote Hrushovski’s non-collapsed ab initio construction for an $n$-ary relation by ${\mathcal{M}}_{\sim ̸}$ and the analogous construction for a symmetric $n$-ary relation by ${\mathcal{M}}_{\sim }$. We show that ${\mathcal{M}}_{\sim ̸}$ is isomorphic to a proper reduct of ${\mathcal{M}}_{\sim }$ and vice versa, and that the combinatorial pregeometries associated with both structures are isomorphic.