Abstract <p>The paper is devoted to determination of quadratic and cubic energy forms for potentials of force and couple stresses of micropolar elastic solids. H/E/A-representations for these potentials have been proposed in earlier discussions. The A-form is a polynomial invariant comprising integer powers of individual and joint base rational algebraic invariants/pseudoinvariants. Some of them are sensitive to mirror reflections and inversions of three-dimensional space. In the present paper, a set of two symmetric and two asymmetric second rank tensors is considered. For these tensors a complete irreducible system of individual and joint cubic integer rational algebraic invariants is obtained. In fact the invariants are invariant traces of inner products of integer powers of tensors belonging to the set. The obtained systems of invariants is then used to determine a cubic approximation of energy form of a hemitropic solid and eventually to specify total 37 constitutive moduli (28 new moduli among them). Based on the obtained results a model of cubic hemitropic micropolar elastic solid is proposed.</p>