  We begin by considering several properties commonly (but not universally)
possessed by B\"acklund transformations between hyperbolic Monge-Amp\`ere
equations: wavelike nature of the underlying equations, preservation of
independent variables, quasilinearity of the transformation, and autonomy of
the transformation. We show that, while these properties all appear to depend
on the formulation of both the underlying PDEs and the B\"acklund
transformation in a particular coordinate system, in fact they all have
intrinsic geometric meaning, independent of any particular choice of local
coordinates.
  Next, we consider the problem of classifying B\"acklund transformations with
these properties. We show that, apart from a family of transformations between
Monge-integrable equations, there exists only a finite-dimensional family of
such transformations, including the well-known family of B\"acklund
transformations for the sine-Gordon equation. The full extent of this family is
not yet determined, but our analysis has uncovered previously unknown
transformations among generalizations of Liouville's equation.
