Currently the practice of constructing various algorithms of inertial navigation, including strapdown navigation system (SINS), indicates that two mathematical methods for describing rotation are mainly used: using quaternions or direction cosines. Moreover, in SINS algorithms it is often convenient to use two of these parameter systems in parallel: quaternions for the angular motion algorithms (the orientation problem), and the direction cosines for the problem of calculating speed and position. In this case, parallel calculation of these two groups of parameters is carried out under the assumption of an exact isomorphic accordance between them. However, if the formalism of quaternions is single-valued, then the apparatus of matrix operations using direction cosines does not possess such a feature and admits several interpretations, which should be borne in mind. From this point of view, the question posed in this article makes sense not only as the question of the existence of an isomorphic accordance (there is no doubt about it), but in what form it exists for the matrix formalism of direction cosines.