There is the following description of $G$ invariant pseudodifferential operators on a Riemannian homogeneous space $G/H$: The Schwartz kernels are smooth outside the diagonal and conormal with respect to the diagonal; this is equivalent to Beals' commutator characterization mentioned in Pedro Lauridsen Ribeiro's answer. Moreover, the full geometric symbols are, modulo symbols of order $-\infty$, invariant under the symplectic action of $G$ on $T^*(G/H)$.
Geometric symbols are defined by pulling back Schwartz kernels under the exponential map of the Levi-Civita connection to a neighbourhood of the zero section of the tangent bundle $T(G/H)$, and then taking Fourier transforms in the fiber variable. See here for more details of and references to the geometric pseudodifferential calculus. That principal symbols must be invariant is clear from the transformation behaviour under diffeomorphisms (special case of Egorov's theorem, or FIO calculus). Lemma 6.2 in the paper in Math. Z., of which I am a coauthor, gives the invariance of the full geometric symbol if the diffeomorphism is an isometry, e.g. an action on $G/H$ by a group element.
The description discards smoothing operators. This is reasonable if one adopts, from Sato's microlocal analysis, the view that pseudodifferential operators operate on microfunctions.
The above characterization of invariant pseudodifferential operators is not algebraic. However, I think that it is simple and straightforward. I don't know if there is a characterization of $G$ invariant microdifferential (pseudodifferential) operators in the setting of algebraic analysis which is more algebraic.