We describe a new family of discrete spaces suitable for use with mixed methods on general quadrilateral and hexahedral elements. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including the hybrid formulation, carries over to these new elements with proofs unchanged. Because transforming general quadrilaterals into squares introduces nonlinearity and because mixed methods involve the divergence operator, the new spaces are more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. These new elements may be useful in topologically regular grids, where initially rectangular grids are deformed to match features of the physical region.