Mathematical reasoning skills are a desired outcome of introductory physics courses, particularly calculus-based courses. Signed quantities are ubiquitous in physics, and sign carries important and varied meanings. Unlike physics experts, novices struggle with the many roles signed numbers can play in physics contexts; recent evidence shows that unresolved struggle carries over to subsequent physics courses. Mathematics education research literature documents cognitive challenges of conceptualizing negative numbers as mathematical objects--for experts, historically, and for novices as they learn. We add to the small but growing body of physics education research that focuses on student reasoning about signed quantities and the role of the negative sign in models. This paper contributes a framework for categorizing the various natures of the negative sign in physics contexts, modeled on the established natures of negativity in algebra from the mathematics education research community. We hope such a framework can facilitate innovation in methods and curricular activities to catalyze a deeper mathematical conceptualization of signed quantities in the introductory courses and beyond.