Abstract
In this work, the two classes of cubic harmonics [F. C. Von der Lage and H. A. Bethe, Phys. Rev. 71, 612 (1947)] which are invariant under permutations of their Cartesian coordinates, are constructed as orthonormal polynomial functions of the latter. Each such function is conveniently expressed as a linear combination of a term of highest degree and other cubic harmonics of lesser degree. All needed expansion coefficients and normalization constants are determined explicitly in terms of the degrees of the functions that are involved.