Let $V$ be a finite dimensional inner product space over $\mathbb{R}$. Let $T$ be a linear operator on $V$. Associated to $T$ is a Jordan/rational decomposition, which we view as a collection of cyclic subspaces $V_1,\ldots,V_r$ such that $V = V_1\oplus V_2\oplus\cdots\oplus V_r$, where each $V_i$ is isomorphic to $\mathbb{R}[t]/f(t)^n$ for some irreducible polynomial $f(t)\in\mathbb{R}[t]$.
What properties on $T$ would allow the existence of such a decomposition where the $V_i$'s are all pairwise orthogonal?
This is true if $T$ is self-adjoint or orthogonal (by various spectral theorems). Is there a nice characterization of $T$'s satisfying this property?