A novel analysis of the neutron multigroup diffusion equation is presented for two-dimensional piecewise homogeneous domains with interior corners that arise at the intersections between regions with distinct material properties. Using polar coordinates centered at a typical interior corner, the solution of the multigroup flux is obtained as an infinite series of products of pairs of functions such that, for every pair, one of the functions depends solely on the angular variable and a single energy group while the other function depends on the radial variable and on all energy groups. The angular functions are shown to be the eigenfunctions of a Sturm-Liouville system that admits an infinite set of discrete and, in general, noninteger eigenvalues. On the other hand, the radial functions are the solutions of an infinite system of second-order ordinary linear differential equations. Exact explicit solutions for the multigroup diffusion equation (MGDE) for two-dimensional disk-like homogeneous domains are also derived and shown to yield analytic expressions for the group fluxes. This analyticity is shown to stem from the fact that the relevant eigenvalues are positive integers, independent of material properties and/or group structure. The exact expressions for the angular eigenvalues and corresponding eigenfunctions for two-region domains are then derived and shown to depend crucially on the specific angle between the two regions. This fact is underscored by deriving the exact expressions for the complete sets of eigenvalues and eigenfunctions for two geometries of particular importance to nuclear reactors, namely the hexagonal and rectangular geometries, respectively, and by showing that they are fundamentally distinct from one another. Of course, these expressions reduce to one and the same form for both geometries when the respective two-region domains are reduced to a single-region domain. Finally, the multigroup fluxes are shown to be bounded but nonanalytic at the respective interior corners; the reason underlying this behavior is traced back to the noninteger character of the relevant eigenvalues. This nonanalyticity is shown to be the fundamental reason for the failure of conventional (e.g., finite difference, finite element) numerical methods for solving the MGDE at and around such corners.