A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new constitutive modeling framework for polyconvex anisotropic hyperelasticity based on triclinic invariants and group symmetrization, and apply it to a PANN model. For finite symmetry groups, this PANN model fulfills all common mechanical constitutive conditions a priori. (2) We propose a new group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We benchmark different polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials. (4) We provide an introduction to the construction of polyconvex invariants, integrity and functional bases and review established polyconvex bases alongside the newly derived ones.