I have heard it mentioned before that $G$-spaces which have equivalent orbit categories must then have equivalent fundamental categories (sometimes called the equivariant fundamental groupoid). This would then give an analogy to the result that homotopic spaces have isomorphic fundamental groups. However I have not been able to find a proof of this theorem, nor even a statement or conjecture.
Given a group $G$, the orbit category $\mathcal{O}(G)$ of $G$ is the category whose objects are sets of left cosets, $G/H$ for $H\leq G$ ($H$ need not be normal - we do not need $G/H$ to be a group itself), endowed with its natural $G$-set structure. An arrow in $\mathcal{O}(G)$ from $G/H$ to $G/K$ is an equivariant map (or $G$-map) $\varphi:G/H\rightarrow G/K$.
Given a $G$-space $X$ (so $X$ is a $G$-set and a topological space, where the map $X\rightarrow X$, $x\mapsto g.x$ is continuous for each $g\in G$), the orbit category $\mathcal{O}(G,X)$ of $G$ over $X$ is the category whose objects are the $G$-maps $\varphi:G/H\rightarrow X$ for arbitrary $H\leq G$. An arrow $\sigma$ from $\varphi:G/H\rightarrow X$ to $\psi:G/K\rightarrow X$ is a $G$-map $\sigma:G/H\rightarrow G/K$ such that $\varphi=\psi\circ\sigma$.
The fundamental category $\Pi(G,X)$ of $G$ over the $G$-space $X$ is the category whose objects are the same as in $\mathcal{O}(G<,X)$ above. An arrow from $\varphi:G/H\rightarrow X$ to $\psi:G/K\rightarrow X$ is a pair $(\sigma,\phi)$, where $\sigma:G/H\rightarrow G/K$ is a $G$-map, and $\phi$ is a homotopy class of paths such that $\phi_0=\varphi$ and $\phi_1=\psi\circ\sigma$. We think of the arrows as being paths in $X$, with the addition that we are allowed to apply a $G$-map at the end of our path. (Let me know if more details are needed for clarity. I am using the definition from tom Dieck's "Transformation Groups".)