We introduce and explore “paired” cosmological simulations. A pair consists of an A and B simulation with initial conditions related by the inversion (underdensities substituted for overdensities and vice versa). We argue that the technique is valuable for improving our understanding of cosmic structure formation. The A and B fields are by definition equally likely draws from initial conditions, and in the linear regime evolve identically up to the overall sign. As nonlinear evolution takes hold, a region that collapses to form a halo in simulation A will tend to expand to create a void in simulation B. Applications include (i) contrasting the growth of A-halos and B-voids to test excursion-set theories of structure formation, (ii) cross-correlating the density field of the A and B universes as a novel test for perturbation theory, and (iii) canceling error terms by averaging power spectra between the two boxes. Generalizations of the method to more elaborate field transformations are suggested.