I have an interior spherical multipole expansion (as in Modern Electrodynamics by Andrew Zangwill): $$f(\textbf{r}):=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} B_{lm} r^{l} Y_{lm}^* $$ with spherical harmonics $Y_{lm}$ and interior spherical multipole moments $B_{lm}$.
I would like to know how the moments change if I flip any of the axes, i.e., going from $\textbf{r}=(x,y,z)$ to $\textbf{r}'=(-x,y,z)$ or analogically for the other two axes. Apparently, only signs of some of the moments (real or imaginary part or both) change but I would like to have some general formula based on $l$ and $m$ for each of the axes.
EDIT:
The goal is to use the moments as ML descriptors. However, my system is oriented into a standard frame using a PCA-like algorithm. As a consequence, the axes are defined up to the sign. Averaging the ML kernel over all 8 possible axes orientations works nicely but I want to make it more efficient and maybe propose another kernel in the end when I understand the symmetries of the moments.