In this paper, we explore the theory of tensor invariants as a mathematical framework for computing new biomarkers for HARDI. We present and explain the integrity basis, basic in- variants and principal invariants of 2nd & 4th order tensors to expand on a recently proposed paper on 4th order tensor invariants. We present the mathematical results and compute the basic and principal invariants on a controlled synthetic dataset and an in vivo human dataset. We show how the in- tegrity bases of these two sets of invariants can form a promis- ing framework for developing new biomarkers for HARDI.