We compute the contributions to continuum photon pair production at
hadron colliders from processes initiated by gluon-gluon and
gluon-quark scattering into two photons through a four-leg virtual
quark loop. Complete two-loop cross sections in perturbative quantum
chromodynamics are combined with contributions from soft parton
radiation resummed to all orders in the strong coupling strength. The
structure of the resummed cross section is examined in detail,
including a new type of collinear singularity affecting azimuthal
angle distributions of photons in the pair's rest frame. As a result
of this analysis, we predict diphoton transverse momentum
distributions in Higgs boson searches at the Tevatron and LHC.