Abstract: In this paper, we present a method for constructing a 3D cross-frame field, a 3D extension of the 2D cross-frame field as applied to surfaces in applications such as quadrangulation and texture synthesis. In contrast to the surface cross-frame field (equivalent to a 4-Way Rotational-Symmetry vector field), symmetry for 3D cross-frame fields cannot be formulated by simple one-parameter 2D rotations in the tangent planes. To address this critical issue, we represent the 3D frames by spherical harmonics, in a manner invariant to combinations of rotations around any axis by multiples of $pi /2$. With such a representation, we can formulate an efficient smoothness measure of the cross-frame field. Through minimization of this measure under certain boundary conditions, we can construct a smooth 3D cross-frame field that is aligned with the surface normal at the boundary. We visualize the resulting cross-frame field through restrictions to the boundary surface, streamline tracing in the volume, and singularities. We also demonstrate the application of the 3D cross-frame field to producing hexahedron-dominant meshes for given volumes, and discuss its potential in high-quality hexahedralization, much as its 2D counterpart has shown in quadrangulation.