This article analyzes the quotient space of compact intervals of R with respect to the family of symmetric intervals. We analyze some algebraic and topological properties of this quotient space. Since the space of compact intervals can be embed on this quotient space, we introduce a concept of differentiability for interval-valued functions and then, we make a comparison with other concepts of differentiability. Finally, we discuss necessary conditions in order to show that the solutions of an interval differential equations considering the first and second type of gH-differentiability belong to the same equivalence class.