We propose a general strategy for enforcing multiple conservation laws and dissipation inequalities in the numerical solution of initial value problems. The key idea is to represent each such structure by means of an associated test function; we introduce auxiliary variables representing the projection of these test functions onto a discrete test set and modify the equation to use these new variables. We may thereby devise novel geometric numerical integrators of arbitrary order for various ODE and PDE systems, including a simultaneously mass-, momentum- and energy-conservative discretisation for the compressible Navier–Stokes equations that provably dissipates entropy, and an integrator for general conservative ODEs that conserves all known invariants. Numerous structure-preserving schemes in the literature further exist as special instances of our framework.