A powerful new idea in the computational representation of structures is that of the digital twin. The concept of the digital twin emerged and developed over the last decade, and has been identified by many industries as a highly desired technology. The current situation is that individual companies often have their own definitions of a digital twin, and no clear consensus has emerged. In particular, there is no current mathematical formulation of a digital twin. A companion paper to the current one will attempt to present the essential components of the desired formulation. One of those components is identified as a rigorous representation theory of models; most importantly, governing how they are verified and validated, and how validation information can be transferred between models. Unlike its companion, which does not attempt detailed specification of any twin components, this paper will attempt to outline a rigorous representation theory of models, based on the introduction of two new concepts: mirrors and virtualizations. The paper is not intended as a passive wish list; it is intended as a rallying call. The new theory will require the active participation of researchers across a number of domains including: pure and applied mathematics, physics, computer science, and engineering. The paper outlines the main objects of the theory and gives examples of the sort of theorems and hypotheses that might be proved in the new framework.