Small- and wide-angle X-ray scattering tensor tomography are powerful methods for studying anisotropic nanostructures in a volume-resolved manner and are becoming increasingly available to users of synchrotron facilities. The analysis of such experiments requires advanced procedures and algorithms, which creates a barrier for the wider adoption of these techniques. Here, in response to this challenge, we introduce the MUMOTT package. It is written in Python, with computationally demanding tasks handled via just-in-time compilation using both CPU and GPU resources. The package has been developed with a focus on usability and extensibility, while achieving a high computational efficiency. Following a short introduction to the common workflow, we review key features, outline the underlying object-oriented framework and demonstrate the computational performance. By developing the MUMOTT package and making it generally available, we hope to lower the threshold for the adoption of tensor tomography and to make these techniques accessible to a larger research community.