  Repulsive and attractive interactions usually lead to very different physics.
Striking exceptions exist in the dynamics of driven-dissipative quantum
systems. For the example of a photonic Bose-Hubbard dimer, we establish a
one-to-one mapping relating the cases of onsite repulsion and attraction. We
prove that the mapping is valid for an entire class of Markovian open quantum
systems with time-reversal invariant Hamiltonian and physically meaningful
inverse-sign Hamiltonian. To underline the broad applicability of the mapping,
we illustrate the one-to-one correspondence between the nonequilibrium dynamics
in a geometrically frustrated spin lattice and that in a non-frustrated partner
lattice.
