  We study the Busemann process of the planar directed polymer model with
i.i.d. weights on the vertices of the planar square lattice, both the general
case and the solvable inverse-gamma case. We demonstrate that the Busemann
process intertwines with an evolution obeying a version of the geometric
Robinson--Schensted--Knuth correspondence. In the inverse-gamma case this
relationship enables an explicit description of the distribution of the
Busemann process: the Busemann function on a nearest-neighbor edge has
independent increments in the direction variable, and its distribution comes
from an inhomogeneous planar Poisson process. Various corollaries follow,
including that each Busemann function has the same countably infinite dense set
of discontinuities in the direction variable. This contrasts with the known
zero-temperature last-passage percolation cases, where the analogous sets are
nowhere dense but have a dense union. The distribution of the asymptotic
competition interface direction of the inverse-gamma polymer is discrete and
supported on the Busemann discontinuities. Further implications follow for the
eternal solutions and the failure of the one force--one solution principle for
the discrete stochastic heat equation solved by the polymer partition function.
