One way to do this is to reduce this to understanding sets of points on the $2$-sphere up to a certain equivalence. If you let $\mathcal{H}_k$ denote the homogeneous polynomials $p$ of degree $k$ on $\mathbb{R}^3$ that are harmonic, i.e., that satisfy $\Delta p = 0$, then $\mathcal{H}_k$ is an irreducible $\mathrm{SO}(3)$-representation of dimension $2k{+}1$. The space $\mathcal{P}_k$ of all homogeneous polynomials of degree $k$ can be written as the direct sum
$$
\mathcal{P}_k = \mathcal{H}_k \oplus (R\cdot\mathcal{H}_{k-2})
\oplus (R^2\cdot\mathcal{H}_{k-4}) \oplus \cdots
$$
where $R = {x_1}^2+{x_2}^2+{x_3}^2$. In particular, there is a canonical 'harmonic part' $H(p)\in \mathcal{H}_k$ for any $p\in \mathcal{P}_k$, and the map $H:\mathcal{P}_k\to \mathcal{H}_k$ is linear. Then one has the following `factorization result':
Proposition: Any $p\in \mathcal{H}_k$ can be 'factored' in the form
$p = H(\ell_1\ell_2\cdots\ell_k)$,
where the 'factors' $\ell_i\in \mathcal{H}_1$ satisfy $|\ell_1| = |\ell_2| = \cdots = |\ell_k|$ and the elements $\ell_i$ are unique up to permutation and replacement of an even number of them by their negatives.
This 'harmonic factorization' is $\mathrm{SO}(3)$-equivariant, so finding normal forms for harmonic polynomials of a fixed degree $k$ reduces to normalizing sets of $k$ points on the $2$-sphere up to permutation and replacing an even number of them by their negatives, which is a combinatorial problem.
The proof of the above proposition goes through the standard representation theory of $\mathrm{SU}(2)$, which is the double cover of $\mathrm{SO}(3)$.
