Consider two finite point sets $\{P_1,P_2,\ldots,P_n\}$ and $\{Q_1,Q_2,\ldots,Q_n\}$ on the plane. Let's say that they are equivalent if there is a bijection that sends $P_i$ to $Q_i$ such that for distinct $i,j,k,l$, the line-segments $P_iP_j$ and $P_kP_l$ intersect if and only if $Q_iQ_j$ and $Q_kQ_l$ also intersect.
Does this equivalence have a name? References would also be helpful. Identifying equivalent planar graph drawings is a motivation.