Inverse scattering (electromagnetic) problems related to cylindrical bodies with arbitrary cross-section were extensively investigated in the literature by assuming that the orientation of the body is known beforehand. This knowledge permits one to formulate the problem as a two-dimensional scalar problem. The case when the orientation of the body is also unknown, which is extremely important from both theoretical and practical points of view, results inevitably in a three-dimensional vector-valued problem. This paper is devoted to the study of a problem of this kind in its rather general form. The analysis is based on a spectral representation of the scattered field and permits one to determine first, under the Born approximation, the orientation. Once the orientation has been determined, the constitutive parameters can then be found by using any one of the already known methods. An illustrative example shows the practical applicability and the accuracy of the theory.