  In this paper, we introduce and study a remarkable class of mechanisms formed
by a $3 \times 3$ arrangement of rigid quadrilateral faces with revolute joints
at the common edges. In contrast to the well-studied Kokotsakis meshes with a
quadrangular base, we do not assume the planarity of the quadrilateral faces.
Our mechanisms are a generalization of Izmestiev's orthodiagonal involutive
type of Kokotsakis meshes formed by planar quadrilateral faces. The importance
of this Izmestiev class is undisputed as it represents the first known flexible
discrete surface -- T-nets -- which has been constructed by Graf and Sauer. Our
algebraic approach yields a complete characterization of all flexible $3 \times
3$ quad meshes of the orthodiagonal involutive type up to some degenerated
cases. It is shown that one has a maximum of 8 degrees of freedom to construct
such mechanisms. This is illustrated by several examples, including cases which
could not be realized using planar faces. We demonstrate the practical
realization of the proposed mechanisms by building a physical prototype using
stainless steel. In contrast to plastic prototype fabrication, we avoid large
tolerances and inherent flexibility.
