The analysis of the classical radial distribution function of a system provides a possible procedure for uncovering interaction rules between individuals out of collective movement patterns. A formal extension of this approach has revealed recently the existence of a universal scaling in the collective spatial patterns of pedestrians, characterized by an effective potential of interaction V(τ) conveniently defined in the space of the times-to-collision τ between the individuals. Here we significantly extend and clarify this idea by exploring numerically the emergence of that scaling for different scenarios. In particular, we compare the results of bidirectional flows when completely different rules of self-avoidance between individuals are assumed (from physical-like repulsive potentials to standard heuristic rules commonly used to reproduce pedestrians dynamics). We prove that all the situations lead to a common scaling in the t-space both in the disordered phase (V(τ) ~ τ− 2) and in the lane-formation regime (V(τ) ~ τ− 1), independent of the nature of the interactions considered. Our results thus suggest that these scalings cannot be interpreted as a proxy for how interactions between pedestrians actually occur, but they rather represent a common feature for bidirectional flows of self-avoiding agents.