  The asymptotic mean value Laplacian - AMV Laplacian - extends the Laplace
operator from $\mathbb{R}^n$ to metric measure spaces through limits of
averaging integrals. The AMV Laplacian is however not a symmetric operator in
general. In this paper therefore a symmetric version of the AMV Laplacian is
considered, and focus lies on when the symmetric and non-symmetric AMV
operators coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds,
we show that they are identical on a large class of metric measure spaces
including locally Ahlfors regular spaces with vanishing metric-measure
boundary. In addition, we study the context of weighted domains of
$\mathbb{R}^n$ where the two operators typically differ, and provide concrete
formulae for these operators also at points where the weight vanishes.
