We present ALMA and ACA [CI]3P1−3P0 (CI) observations of NGC6240, which we combine with ALMA CO(2-1) and IRAM PdBI CO(1-0) data to study the physical properties of the massive molecular (H2) outflow. We discover that the receding and approaching sides of the H2 outflow, aligned east-west, exceed 10 kpc in their total extent. High resolution (0.24") CI line images surprisingly reveal that the outflow emission peaks between the two AGNs, rather than on either of the two, and that it dominates the velocity field in this nuclear region. We combine the CI and CO(1-0) data to constrain the CO-to-H2 conversion factor (αCO) in the outflow, which is on average 2.1±1.2 M⊙(K km s−1 pc2)−1. We estimate that 60±20 % of the total H2 gas reservoir of NGC6240 is entrained in the outflow, for a resulting mass-loss rate of M˙out=2500±1200 M⊙ yr−1 ≡50±30 SFR. This energetics rules out a solely star formation-driven wind, but the puzzling morphology challenges a classic radiative-mode AGN feedback scenario. For the quiescent gas we compute ⟨αCO⟩=3.2±1.8 M⊙(K km s−1 pc2)−1, which is at least twice the value commonly employed for (U)LIRGs. We observe a tentative trend of increasing r21≡LCO(2−1)′/LCO(1−0)′ ratios with velocity dispersion and measure r21>1 in the outflow, whereas r21≃1 in the quiescent gas. We propose that molecular outflows are the location of the warmer, strongly unbound phase that partially reduces the opacity of the CO lines in (U)LIRGs, hence driving down their global αCO and increasing their r21 values.