We study the Busemann process and competition interfaces of the planar directed polymer model with i.i.d.\ weights on the vertices of the planar square lattice, in both the general case and the solvable inverse-gamma case. We prove new regularity properties of the Busemann process without reliance on unproved assumptions on the shape function. For example, each nearest-neighbor Busemann function is strictly monotone and has the same random set of discontinuities in the direction variable. When all Busemann functions on a horizontal line are viewed together, the Busemann process intertwines with an evolution that obeys a version of the geometric Robinson-Schensted-Knuth correspondence. When specialized to the inverse-gamma case, this relationship enables an explicit distributional description: 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. The distribution of the asymptotic competition interface direction of the inverse-gamma polymer is discrete and supported on the Busemann discontinuities which -- unlike in zero-temperature last-passage percolation -- are dense. Further implications follow for the eternal solutions and the failure of the one force -- one solution principle of the discrete stochastic heat equation solved by the polymer partition function.