In this paper we introduce and study a remarkable class of mechanisms formed by a 3 × 3 arrangement of rigid and skew quadrilateral faces with revolute joints at the common edges. These Kokotsakis-type mechanisms with a quadrangular base and non-planar faces are a generalization of Izmestiev's orthodiagonal involutive type of Kokotsakis polyhedra formed by planar quadrilateral faces. Our algebraic approach yields a complete characterization of all complexes of the orthodiagonal involutive type. It is shown that one has 8 degrees of freedom to construct such mechanisms. This is illustrated by several examples, including cases that are not possible with planar faces.