We theoretically analyze the dynamical evolution of photonic quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics. Our flexible design allows discrete observations of continuous-time quantum walks of photons in a variety of waveguide arrays. Furthermore, our design allows one to inject photons during the evolution, allowing the possibility of interacting with the photons as they are "walking." We find that nontrivial array topologies introduce time-dependent symmetries of the two-photon correlations. These properties allow a large degree of control for quantum state engineering of multimode entangled states in these devices.