Bivariate interpolatory Hermite subdivision schemes are recently applied to build free-form subdivision surfaces. It is well-known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces -- surfaces with unwanted wiggles or undulations -- and non-interpolatory (a.k.a. {\it approximating} in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we construct, using the general theory of vector refinement equation, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We study also symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.