  Inviscid invariants of flow equations are crucial in determining the
direction of the turbulent energy cascade. In this work we investigate a
variant of the three dimensional Navier-Stokes equations that shares exactly
the same ideal invariants (energy and helicity) and the same symmetries (under
rotations, reflexions and scale transforms) as the original equations. It is
demonstrated that the examined system displays a change in the direction of the
energy cascade when varying the value of a free parameter which controls the
relative weights of the triadic interactions between different helical Fourier
modes. The transition from a forward to inverse cascade is shown to occur at a
critical point in a discontinuous manner with diverging fluctuations close to
criticality. Our work thus supports the observation that purely isotropic and
three dimensional flow configurations can support inverse energy transfer when
interactions are altered and that inside all turbulent flows there is a
competition among forward and backward transfer mechanism which might lead to
multiple energy-containing turbulent states.
