We introduce and develop a novel particle exchange Monte Carlo method. Whereas existing methods apply to eigenfunction problems where the eigenvalue is known (e.g., integrals with respect to a Gibbs measure, which can be interpreted as corresponding to eigenvalue zero), here the focus is on problems where the eigenvalue is not known a priori. To obtain an appropriate particle exchange rule we must consider a pair of processes, with one evolving forward in time and the other backward. Applications to eigenfunction problems corresponding to quasistationary distributions and ergodic stochastic control are discussed.