The construction of suitably regular finite element spaces is a crucial ingredient in obtaining stable and well-posed discretisations of mixed formulations of partial differential equations, e.g. the Hellinger–Reissner formulation of linear elasticity. In this work we present a unified approach for constructing the basis functions of partially continuous tensor-valued finite elements including the Regge, Hellan-Herrmann-Johnson, Schöberl-Pechstein [2], Hu-Zhang and Hu-Ma-Sun elements. We also introduce a new tensor finite element, which is minimally H(Div)-conforming and quasi-symmetric. In our approach the basis functions are constructed on the reference cell from template tensors associated with geometric polytopes of the cell and any scalar-valued H^1-conforming finite element space [3]. The tensor-valued finite element basis functions are then built directly via the composition of the polytopal template tensors and the basis functions of the underlying H^1-conforming finite element space. In the case of the Regge, Hellan-Hermann-Johnson and Schöberl-Pechstein elements, the basis functions are then pushed forward to the physical cell in the mesh via standard double Piola mappings. For the Hu-Zhang element, the Hu-Ma-Sun element and our new element, the consistent transformations are not Piola-type and represent an additional novelty of the polytopal approach [1]. We prove a linearly independent construction of the basis on triangular and tetrahedral meshes and demonstrate optimal order convergence via numerical benchmarks.