The thermodynamics and mechanics of non-isothermal polymeric fluids are examined within the auspices of a new methodology wherein the laws of physics and principles of mechanics which are applicable to these thermodynamic systems are imbedded in a definite mathematical structure of a general, abstract equation. Such a concept allows new insight to be obtained concerning some aspects of non-isothermal flows of polymeric fluids, and permits a consistent expression and interpretation of other thermodynamic theories for these systems which have been developed over the past forty years. A major portion of this article is devoted to demonstrating the above statements, and in so doing some common misconceptions occurring in a significant fraction of the literature regarding this subject are exposed. The definite mathematical structure of the new methodology permits the thermodynamically consistent generalization of isothermal, incompressible models of polymeric fluids to non-isothermal, compressible conditions. Doing thus reproduces, corrects, and extends non-isothermal models which have been developed over the years, and also allows for simpler (but equivalent) representations of these models in terms of alternate variables with a clearer connection to the microstructure of the material than the stress tensor and heat flux vector fields. Furthermore, a generalization of the GENERIC structure is proposed that accommodates interactions between phenomena of differing parities, which impose antisymmetry upon the corresponding elements of the dissipative operator matrix.