The fluorescence polarization intensities from fluorescent probes and the electron paramagnetic resonance spectra from spin probes, specifically modifying elements of a biological assembly such as myosin sulfhydryl 1 (SH1) in muscle fibers, are interpreted in terms of probe order parameters using a model-independent method. The probe order parameters are related to each other by an Euler rotation of coordinates. We use this relationship to link the sets of order parameters from the different probes and in so doing create a system of equations that can be solved using only the information available from the experimental data. The solution yields the Euler angles relating the different probe coordinate frames and a larger set of probe order parameters than can be directly detected experimentally. The Euler angles are used to display the relative orientation of the probe molecular frames. The order parameters give rise to probe angular distributions that are at the theoretical limit of resolution. We demonstrate the utility of this analytical method by investigating the rotation of myosin SH1 from its orientation in rigor upon the binding of the nucleotide MgADP to the myosin cross-bridge. Our findings, discussed in the accompanying paper, suggest that the rigor-to-MgADP cross-bridge angular transition consists predominantly of a rotation about the hydrodynamic axis of symmetry of the cross-bridge, i.e., its torsional degree of freedom [Ajtai, K., Ringler, A., & Burghardt, T. P. (1992) Biochemistry (following paper in this issue)].