  We present a novel and remarkably simple formulation of degenerate
higher-order scalar-tensor (DHOST) theories whose Lagrangian is quadratic in
second derivatives of some scalar field. Using disformal transformations of the
metric, we identify a special "frame" (or metric) for which the Lagrangian of
quadratic DHOST theories reduces to the usual Einstein-Hilbert term plus a few
terms that depend on simple geometric quantities characterizing the uniform
scalar field hypersurfaces. In particular, for quadratic DHOST theories in the
physically interesting class Ia, the Lagrangian simply consists of the
Einstein-Hilbert term plus a term proportional to the three-dimensional scalar
curvature of the uniform scalar field hypersurfaces. The classification of all
quadratic DHOST theories becomes particularly transparent in this geometric
reformulation, which also applies to scalar-tensor theories that are degenerate
only in the unitary gauge.
