Using the FEEC (finite element exterior calculus) framework [2], we propose a new formulation of the auto-advection term in the Navier-Stokes equation, using only a gradient operator and an interior product corresponding to the scalar product. With this novel operator, we propose a discretization of the system which is proven to conserve the mass, energy and momentum. Following [1] we then propose a conga (conforming/nonconforming Galerkin) scheme to improve the performances of the solver. A jump penalization is added to stabilize the scheme and the discretization is proven to remain mass and momentum preserving, while the energy is dissipated through the jumps. Both conforming and non-conforming discretization are implemented using splines spaces [3] and preservation properties are confirmed by the simulations. High order convergence is demonstrated for the conforming method, while a new moment-preserving conforming projection needs to be introduced in order to recover high order in the broken case. Long time stability is then studied, showing high dependence on the jump penalization to recover the same accuracy as in the conforming case.