Differential and Linear trail propagation in Noekeon
This is due to the duality between linear and differential trails.
Let $L$ be an invertible linear map on $\mathbb{F}_2^n$, think of it as a matrix for convenience.
In general, a nonzero differential $\Delta_1 \to \Delta_2$ over $L$ must satisfy

$$\Delta_2 = L\,\Delta_1.$$

A nonzero linear approximation $u_1 \to u_2$, however, must satisfy

$$u_2 = L^{-\top}\,u_1$$

An elementary way to see this is to observe that $u_1^\top x = u_2^\top (Lx)$ is equivalent to $u_1^\top x = (L^\top\,u_2)^\top x$. This holds for all $x \in \mathbb{F}_2^n$ whenever $u_2 = L^{-\top}\,u_1$, and otherwise for half (some hyperplane) the $x$.

If $L$ is orthogonal, then $L^{-T} = L$. So then we have both $\Delta_2 = L\Delta_1$ and $u_2 = L u_1$.