The coefficients of mutual coupling of rectangular dielectric resonators in open space are calculated under the condition of their rotation relative to one of the axes of a given rectangular coordinate system. Analytical formulas for complex coupling coefficients are obtained. The expressions found give complete information about the frequencies and Q-factor of coupled oscillations of dielectric resonators. The dependences of the coupling coefficients on the angles of rotation and spatial coordinates of resonators in the case of excitation of the main magnetic types of natural oscillations in them are considered. The concept of pseudo-rotation of resonators is introduced. Cases are noted when the pseudo-rotation of the resonators does not lead to a change in the coupling coefficients. The dependences of the coupling coefficients for different types of resonator pseudo-rotations are investigated. New integral representations are derived for the mutual coupling coefficients of rectangular dielectric resonators provided that their axes rotate in open space. In particular cases of parallelism of the resonator axes of one of the coordinate axes, the analytical expressions found in the work coincide with those obtained earlier. For each case of rotation, approximate analytical formulas are found for the integral representations obtained in this work, expressed in terms of the spherical Hankel functions of the second kind. Comparison of calculations of coupling coefficients by integral formulas and approximate expressions is carried out. It is shown that the approximate expressions have acceptable accuracy for all the considered cases of rotations. The dependences of the coupling coefficients on the coordinates of the resonators are investigated. The regions are marked in which the found integral representations make it possible to correctly describe the coupling coefficients of rectangular resonators. In contrast to integral representations, approximate formulas are correct in the entire spatial region of resonator interaction. The results obtained make it possible to construct analytical models of antennas, multi-element arrays and devices of infrared and optical wavelength ranges, made with the use of rectangular dielectric resonators; significantly reduce computation timecompared to numerical methods and optimize complex multi-cavity structures of microwave and optical communication systems.