Let $G$ be a compact Lie group that acts on a smooth, finite dimensional, oriented manifold $M$, and suppose that such action preserves orientation, i.e., for each $g\in G$, the diffeomorphism $\mu_g$ of $M$ induced by the action preserves orientation. Consider the stratification $$M=\bigcup M_j$$ by orbit types, i.e., $M_j$ is the set of points that have the same isotropy, up to conjugation.

My question is, are there "simple" conditions under which the quotient manifold $M_j/G$ is orientable? Is it automatic true with the orientation hypothesis on $M$ and the action?