In this talk we will present and discuss some new results regarding the existence of hitherto unknown relations that exist between certain Bernstein diffusions on the one hand, and processes which typically occur in forward-backward systems of stochastic differential equations on the other hand. More specifically, the Bernstein diffusions we will consider can wander in bounded convex domains of Euclidean space, and are generated there by a forward-backward system of decoupled linear deterministic parabolic partial differential equations. This makes them reversible Itô diffusions under some conditions that pertain to their marginal distributions, which then allows the construction of processes which are weak solutions to suitably defined forward-backward systems of coupled stochastic differential equations. This is joint work with Ana Bela Cruzeiro.