A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.