The attractive tail of the intermolecular interaction affects very weakly the structural properties of liquids, while it affects dramatically their dynamical ones. Via the numerical simulations of model systems not prone to crystallization, both in three and in two spatial dimensions, here we demonstrate that the non-perturbative dynamical effects of the attractive forces are tantamount to a rescaling of the activation energy by the glass transition temperature Tg: systems only differing in their attractive interaction have the same structural and dynamical properties if compared at the same value of T /Tg.According to the 'van der Waals picture' the physics of liquids is dominated by the harsh and short-ranged repulsive forces between the particles, the weaker and longer ranged attractive forces only providing a homogeneous cohesive background. This suggests the possibility of treating the attractive forces perturbatively, as first proposed by Weeks, Chandler and Andersen [1,2]. They indeed considered that, due the smooth spatial dependence of the attractive forces and the roughly homogeneous liquid structure, the sum of attractive forces experienced by a particle may be negligible with respect to the sum of the attractive ones. Hence 'the arrangements and motions of molecules . . . are determined primarily by the local packing and steric effects produced by the repulsive forces' [2]. Berthier and Tarjus [3,4] investigated the validity of this scenario focusing on the Kob-Andersen binary Lennard-Jones (KA-LJ) model [5], a prototypical glass former. To asses the relevance of the attractive forces, they compared this model with its WCA variant (KA-WCA), where particles interact via the purely repulsive potential obtained by truncating the LJ potential at its minimum. Their results demonstrated that the attractive forces have a non-perturbative effect on the relaxation dynamics, as the attractive forces greatly slow down the dynamics at low temperatures. Subsequent works have clarified that the difference between attractive and purely repulsive interactions could be attributed to the small structural differences induced by the attractive forces. These differences have been first identified in higher order structural correlations [6-9] and more recently, investigating two-point correlation functions via machine learning techniques [10]. Relating the relaxation time to the configurational entropy through the Adam-Gibbs relation [11], the effect of the different pair correlations on the dynamics has been rationalized considering their different contribution to the entropy [12][13][14]. It has also been demonstrated that it is possible to design a purely-repulsive potential that seemingly generates many structural and dynamical properties of LJ liquids [15][16][17].All of these results indicate that in molecular liquids the attractive forces do not have a perturbative effect as originally speculated. The open question ahead is therefore quantitatively rationalizing their non-perturbative influence. In this Letter, we c...