People care about decision outcomes and how decisions get made, both when making decisions and reflecting on decisions. But formalizing the full range of normative concerns that drive decisions is an open challenge. We introduce Axiomatic Choice as a framework for making and evaluating decisions based on formal normative statements about decisions. These statements, or axioms, capture a wide array of desiderata, e.g., ethical constraints, beyond the typical treatment in Social Choice. Using our model of axioms and decisions we define key properties and introduce a taxonomy of axioms which may be of general interest. We then use these properties and our taxonomy to define the Decision-Evaluation Paradox, formalize the concepts of transparency and deception in explaining and justifying decisions, and reveal the limits of existing methods using axioms to make decisions.