Subdivision schemes are iterative methods for the construction of curves and surfaces from a set of initial control points. Here we focus on the univariate case (i.e. curves) and on interpolatory subdivision (i.e. schemes with a limit curve that interpolates the initial data). This family of schemes can be subdivided in two classes, primal and dual. An algebraic characterization of both classes in terms of certain Laurent polynomials (called symbols) will be provided, together with some examples.