  There are two main types of rank 2 B\"acklund transformations relating a pair
of hyperbolic Monge-Amp\`ere systems, which we call Type $\mathscr{A}$ and Type
$\mathscr{B}$. For Type $\mathscr{A}$, we completely determine a subclass whose
local invariants satisfy a specific but simple algebraic constraint; such
B\"acklund transformations are parametrized by a finite number of constants,
whose cohomogeneity can be either 2, 3 or 4. In addition, we present an
invariantly formulated condition that determines whether a generic Type
$\mathscr{B}$ B\"acklund transformation is one that, under suitable choices of
local coordinates, relates solutions of two PDEs of the form $z_{xy} =
F(x,y,z,z_x,z_y)$ and preserves the $x,y$ variables on solutions.
