We investigate numerically the model proposed in Sahoo et al. (2017 Phys. Rev. Lett. 118 , 164501) where a parameter λ is introduced in the Navier–Stokes equations such that the weight of homochiral to heterochiral interactions is varied while preserving all original scaling symmetries and inviscid invariants. Decreasing the value of λ leads to a change in the direction of the energy cascade at a critical value λ c ∼ 0.3 . In this work, we perform numerical simulations at varying λ in the forward energy cascade range and at changing the Reynolds number R e . We show that for a fixed injection rate, as λ → λ c , the kinetic energy diverges with a scaling law E ∝ ( λ − λ c ) − 2 / 3 . The energy spectrum is shown to display a larger bottleneck as λ is decreased. The forward heterochiral flux and the inverse homochiral flux both increase in amplitude as λ c is approached while keeping their difference fixed and equal to the injection rate. As a result, very close to λ c a stationary state is reached where the two opposite fluxes are of much higher amplitude than the mean flux and large fluctuations are observed. Furthermore, we show that intermittency as λ c is approached is reduced. The possibility of obtaining a statistical description of regular Navier–Stokes turbulence as an expansion around this newly found critical point is discussed. This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.