Zero-dimensional $\sigma$-homogeneous spaces

All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is $\sigma$-homogeneous. Inspired by this theorem, we obtain the following results: assuming $\mathsf{AD}$, every zero-dimensional space is $\sigma$-homogeneous; assuming $\mathsf{AC}$, there exists a zero-dimensional space that is not $\sigma$-homogeneous; assuming $\mathsf{V=L}$, there exists a coanalytic zero-dimensional space that is not $\sigma$-homogeneous. Along the way, we introduce two notions of hereditary rigidity, and give alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is $\sigma$-homogeneous.