  An operator that generates an approximate symmetry of long-range Rydberg
molecules (LRRMs) formed by two alkali atoms, one in a Rydberg state and the
other in the ground state, is identified. This is first done by evaluating the
natural orbitals associated to a variational calculation of the binding wave
function within the Born-Oppenheimer description of the molecule including $s-$
and $p-$ Fermi pseudopotential and the hyperfine structure energy terms. The
resulting orbitals with highest occupation number are shown to be identical to
those obtained by a perturbative model for high angular momentum -- trilobite
and butterfly -- LRRMs. Whenever the slight dependence of the quantum defects
of the Rydberg electron on its total momentum $\vec j = \vec \ell +\vec s_1$
can be neglected, the symmetry operator of the high angular momentum LRRMs
orbitals is identified as the sum of the spin of the Rydberg electron $\vec
s_1$, spin of the valence electron $\vec s_2$ and the spin of nucleus $\vec i$
of the ground state atom, $\vec {N} =\vec {s_1} + \vec {s_2} + \vec {i}$. The
spin-orbitals that diagonize $\vec{N}$ define compact basis sets for the
description of LRRMs beyond the aforementioned approximations. The matrix
elements of the Hamitonian in these basis sets have simple expressions, so that
the relevance of triplet and singlet contributions can be directly estimated.
The expected consequences of this approximate spin-symmetry on the spectra of
LRRMs are briefly described.
