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

Geometric characterization and classification of B\"acklund transformations of sine-Gordon type