  We introduce and explore "paired" cosmological simulations. A pair consists
of an A and B simulation with initial conditions related by the inversion
$\delta_A(x, t_{initial})=-\delta_B(x,t_{initial})$ (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 {\Lambda}CDM initial conditions,
and in the linear regime evolve identically up to the overall sign. As
non-linear 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.
