The aim of this work is to derive formulas for numerical calculations of the orientation-dependent London van der Waals (vdW) interaction energy (V-A) between two rectangular bodies with arbitrary dimensions, arranged at arbitrary relative angles (theta) and separations in twisted and coplanar rotational modes. The formulation is made using a simple volume-element-integration method in the framework of the microscopic approach, in which V-A is the sum of the local vdW energy (V-p) between body 1 and each thin plate constituting body 2. Examples of the calculation results are the following: (1) The theta values that give maximal and minimal values of V-A depend on their shapes and relative positions. (2) As the bodies come close to each other, the variations of V-A with theta and thus vdW dispersion torques generated are drastically intensified. (3) Upon increasing the length of crossing rods in twisted configurations, the V-A values become constant beyond a critical length (depending on theta and separation), where the length effect on V-A disappears. (4) The distribution curves of V-p show that the region in body 2 which interacts effectively with body 1 (i.e., the effective interaction region) is more sharply localized in the vicinity of the surface (closest to body 1) as the separation is decreased.