Let $V$ be a finite-dimensional real inner product space and let $\tau$ be an endomorphism of $V$. Let $V=V_1 \bigoplus \cdots \bigoplus V_r$ be the decomposition of $V$ into $\tau$-invariant and $\tau$-cyclic subspaces, corresponding to the elementary divisor decomposition of $\tau$. I know that if $\tau$ is normal with respect to the inner product, then $V_i \perp V_j$ $\forall i \neq j$. Do there exist any more general conditions on $\tau$, i.e. less strong than normality, such that the subspaces $V_i$ are orthogonal?