Electron-electron-nuclear three-spin mixing is discussed as a source of a new type of CIDNP effects that is peculiar to the solid state. The build-up of nonequilibrium nuclear spin polarization by this mechanism does not require singlet-tripler branching with different fates of the radicals in the two branches. Rather it is a polarization transfer from the electron-electron zero-quantum transition to the nuclear spins in the vicinity of a 2-fold avoided level crossing. An analytical expression is derived that describes this transfer for short times after the radical generation. Under this condition, the size of the effect is proportional to the static field, to the electron-electron spin coupling, and to the square of the anisotropy of the hyperfine coupling. An estimate of the order of magnitude of the resulting polarization reveals that three-spin mixing could be the source of recently observed solid-state CIDNP effects in bacterial photosynthetic reaction centers. CIDNP effects of this type should occur in a sizeable number of photochemical or thermal reactions in organic solid-state chemistry that involve radical pairs as intermediates.