For prescribed values of a function and its partial derivatives of orders $1$ and $2$ at the vertices of a square, we fit an interpolating surface. We investigate two families of solutions provided by two Hermite subdivision schemes, denoted $HD^2$ and $HR^2$. Both schemes depend on 2 matrix parameters, a square matrix of order 2 and a square matrix of order 3. We exhibit the masks of both schemes. We compute the Sobolev smoothness exponent of the general solution of the Hermite problem for the most interesting schemes $HD^2$ and $HR^2$ and we get a lower bound for the H\"{o}lder smoothness exponent. We generate a $C^2$ interpolant on any semiregular rectangular mesh with Hermite data of degree 2.