  We revisit the moving least squares (MLS) approximation scheme on the sphere
$\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using
the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields
for functions in $\mathcal C^{L+1}(\mathbb S^{d-1})$ the approximation order
$\mathcal O \left( h^{L+1} \right)$, where $h$ denotes the fill distance of the
sampling nodes.
  In this paper we show that the dimension of the ansatz space can be almost
halved, by including only spherical harmonics of even or odd degree up to $L$,
while preserving the same order of approximation. Numerical experiments
indicate that using the reduced ansatz space is essential to ensure the
numerical stability of the MLS approximation scheme as $h \to 0$. Finally, we
compare our approach with an MLS approximation scheme that uses polynomials on
the tangent space as ansatz space.
