We consider the interaction between nematic liquid crystals (NLCs) and polymer substrates. Such substrates can interact with NLCs, exhibiting a phenomenon known as director gliding: the preferred orientation of the NLC molecules at the interface changes on time scales that are slow relative to the elastic relaxation time scale of the NLC. We present two models for gliding, inspired by experiments that investigate the interaction between the NLC and a polymer substrate. These models, though simple, lead to nontrivial results, including loss of bistability under gliding. Perhaps surprisingly, we find that externally imposed switching between the steady states of a bistable system may reverse the effect of gliding, preventing loss of bistability if switching is sufficiently frequent. Our findings may be of relevance to a variety of technological applications involving liquid crystal devices, and particularly to a new generation of flexible liquid crystal displays that implement polymeric substrates.