Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal group O(d).
Consider the operator $\mathcal{M}_1 \to \mathcal{M}_1$ which takes a measure $\pi \in \mathcal{M}_1$ to $\overline{\pi}$, which satisfies $$ \overline \pi (E) = \int_{O(d)} \pi(QE) \, \mathrm{d} \sigma(Q), $$ for any measurable subset $E$ of the unit ball. Note that we define $QE = \{Q x: x\in E\}$. Clearly the mapping $\pi \mapsto \overline \pi$ produces $O(d)$-invariant measures.
Is this mapping well known? It is similar to the concept of "symmetrization" of scalar random variables, in the sense that a random variable sampled from $\overline \pi$ has the same distribution as $Q \theta$ where $Q$ is uniform on $O(d)$ and $\theta$ is (independently) drawn from $\pi$.
Some concrete questions are: is it a surjection to the family of rotationally invariant probability measures on $\Theta$? Is it invertible on its image? What else is known about it?
