We discretize the incompressible Navier–Stokes equations on a polytopal mesh by using mimetic reconstruction operators. The resulting method conserves discrete mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. To do this we introduce a dual mesh and show how the dual mesh can be completed to a cell-complex. We present existing mimetic reconstruction operators in a new symmetric way applicable to arbitrary dimension, use these to interpolate between primal and dual mesh and derive properties of these operators. Finally, we test both 2- and 3-dimensional versions of the method on a variety of complicated meshes to show its wide applicability. We numerically test the convergence of the method and verify the derived conservation statements.