In this paper, we present a new class of interpolatory Hermite subdivision schemes of order 2 reproducing polynomials. Each member in this class, denoted by Hn for n ≥ 1, preserves polynomials of degree up to 4n + 1 admitting the approximation order of 4n + 2. Furthermore, it has free parameters which provide flexibility in designing curves/surfaces. H1, the simplest and the most attractive scheme in this class, achieves C4 smoothness with the parameters in certain ranges, and its performance is demonstrated with numerical examples.