There is an analogous situation in quantum mechanics. It is of course well known that the finite-dimensional irreducible complex projective representations of $SO(3)$ are parametrized by a non-negative half-integer $s=0,1/2,1,3/2,\ldots$ with dimension $2s+1$. However, on these representations the projective action of $SO(3)$ is only transitive if $s=0$ or $s=1/2$. Understanding the orbits of $SO(3)$ on these representations is thus a question of interest.
A simple solution was found by Majorana, commonly known as the Majorana stellar representation. A general nonzero vector in the spin $s$ representation can be written as the symmetrized tensor product of $2s$ nonzero vectors in the spin-$1/2$ representation. This is unique up to scaling the vectors by factors which multiply to $1$ and by reordering the vectors. Passing to the projective spaces, a general state is represented by a collection of $2n$ unlabeled (not necessarily distinct) points on the <PERSON> sphere. The action of $SO(3)$ here is just rotations of the sphere.
Thus a dense subset of the projective space $P^{2s}$ is mapped to the nondegenerate configurations of $2s$ unlabelled points on a sphere, which is a well-studied example of a configuration space. The topology of configuration spaces is well-understood by mathematicians. 364
For example, its fundamental group is the quotient of the braid group on $2s$ strands by a single relator (see this paper for a proof). That configuration space inherits the $SO(3)$ action and the orbit space is the quotient of that action. There are also degenerate orbits when $2$ or more of the points coincide, and these prevent the orbit space from being a legitimate manifold (it is, however, an orbifold).
To relate this back to multipole distributions, actually we don't really need to do much work at all. The $Y_l^m$ spherical harmonics for $m=-l,\ldots,l$ span a copy of the $s=l$ representation of $SO(3)$. Here we are looking at a real representation rather than a complex representation. As such we must restrict the space to the appropriate real section spanned by real spherical harmonics. Additionally, in this case it is an ordinary representation, not a projective representation. This means that we get $1$ degree of freedom back from rescaling by real numbers (rescaling by complex numbers is disallowed). Finally, the quotient by the $SO(3)$ action kills 3 real degrees of freedom for sufficiently large $l$ such that the action is faithful (in this case already $l=2$ is sufficient). In particular then, the dimensionality of the orbit space is, for $l=0,1,2,3,4,\ldots$, $$d_l=1,1,2,4,6,\ldots.$$ 976