Does magnetic moment change under inversion symmetry?

Since magnetic moment can be view as a small electric current circle. Pictorially, when apply inversion operation, the current direction is reversed, so I think the $$\vec{m}\to -\vec{m}$$ under inversion symmetry operation.

On the other hand, the formula for the magnetic moment is $$\vec{m}=\int_V \vec{r}\times\vec{j}\mathrm{d}V$$. Under the inversion symmetry operation $$\vec{r}\to -\vec{r}$$ and $$\vec{j}\to -\vec{j}$$, therefore $$\vec{m}$$ is unchanged.

The above two reasoning must have one being wrong, which one and what is the flaw of the reasoning?

• Which inversion symmetry? – Qmechanic Jul 31 at 15:53
• @Qmechanic I think any inversion should be same for a circular loop current – an offer can't refuse Aug 1 at 5:40

$$\vec m$$ and $$\vec B$$ are indeed "axial vectors" and invariant under space inversion. Imagine a circular current. Under inversion the current changes direction but also the circle is inverted. This leaves $$\vec m$$ unchanged. (just noticed Ben Crowell's ear lier answer which is identical to mine.
$$\vec E$$ also is not a true 3D vector as it changes sign under time reversal.