  We completely describe by inequalities the set of boundary correlation
matrices of planar Ising networks embedded in a disk. Specifically, we build on
a recent result of M.~Lis to give a simple bijection between such correlation
matrices and points in the totally nonnegative part of the orthogonal
Grassmannian, which has been introduced in 2013 in the study of the scattering
amplitudes of ABJM theory. We also show that the edge parameters of the Ising
model for reduced networks can be uniquely recovered from boundary
correlations, solving the inverse problem. Under our correspondence, the
Kramers--Wannier high/low temperature duality transforms into the cyclic
symmetry of the Grassmannian, and using this cyclic symmetry, we prove that the
spaces under consideration are homeomorphic to closed balls.
