  In this paper we study the dimension of splines of mixed smoothness on
axis-aligned T-meshes. This is the setting when different orders of smoothness
are required across the edges of the mesh. Given a spline space whose dimension
is independent of its T-mesh's geometric embedding, we present constructive and
sufficient conditions that ensure that the smoothness across a subset of the
mesh edges can be reduced while maintaining stability of the dimension. The
conditions have a simple geometric interpretation. Examples are presented to
show the applicability of the results on both hierarchical and non-hierarchical
T-meshes. For hierarchal T-meshes it is shown that mixed smoothness spline
spaces that contain the space of PHT-splines (Deng et al., 2008) always have
stable dimension.
