We develop an operator-based description of two types of multimode-entangled single-neutron quantum optical devices: Wollaston prisms and radio-frequency spin flippers in inclined magnetic-field gradients. This treatment is similar to the approach used in quantum optics, and is convenient for the analysis of quantum contextuality measurements in certain types of neutron interferometers. We describe operationally the way multimode-entangled single-neutron states evolve in these devices, and provide expressions for the associated operators describing the dynamics, in the limit in which the neutron state space is approximated by a finite tensor product of distinguishable subsystems. We design entangled-neutron interferometers to measure entanglement witnesses for the Clauser, Horne, Shimony, and Holt and Mermin inequalities, and compare the theoretical predictions with recent experimental results. We present the generalization of these expressions to entangled distinguishable subsystems, which could become relevant in the future if it becomes possible to add neutron orbital angular momentum to the experimentally accessible list of entangled modes. We view this paper as a necessary first step towards a theoretical description of entangled neutron scattering from strongly entangled matter, and we explain why it should be possible to formulate a useful generalization of the usual Van Hove linear-response theory for this case. We also briefly describe some other scientific extensions and applications which can benefit from interferometric measurements using the types of single-neutron multimode entanglement described by this analysis.