This article studies rank 2 Bäcklund transformations of hyperbolic Monge-Ampère systems using Cartan's method of equivalence. Such Bäcklund transformations have two main types, which we call Type A and Type B. For Type A, we completely determine a subclass whose local invariants satisfy a specific but simple algebraic constraint. We show that such Bäcklund transformations are parametrized by a finite number of constants; in a subcase of maximal symmetry, we determine the coordinate form of the underlying PDEs, which turn out to be Darboux integrable. For Type B, we present an invariantly formulated condition that determines whether a Bäcklund transformation is one that, under suitable choices of local coordinates, relates solutions of two PDEs of the form zxy=F(x,y,z,zx,zy) and preserves the x,y variables on solutions.