= Solution
Partition a regular parameter into interest $\psi$ and nuisance $\lambda$. <Parameter orthogonality> means the cross-block of the expected <Fisher information matrix> vanishes:
$$
\boxed{I_{\psi\lambda}=\mathbb E[U_\psi U_\lambda^\top]=-\mathbb E\ell_{\psi\lambda}=0.}
$$
It is expected-information orthogonality, not a demand that every observed cross derivative be zero. To first order, the asymptotic normal distribution of the joint <maximum-likelihood estimator> has independent interest and nuisance blocks. Its efficient interest information simplifies from the Schur complement $I_{\psi\psi}-I_{\psi\lambda}I_{\lambda\lambda}^{-1}I_{\lambda\psi}$ to $I_{\psi\psi}$.
For scalar interest, new nuisance coordinates $\eta$ can locally be chosen by the <local orthogonal nuisance reparametrization>
$$
\partial_\psi\lambda(\psi,\eta)=-I_{\lambda\lambda}^{-1}I_{\lambda\psi}.
$$
Indeed the new interest score is $U_\psi+(\partial_\psi\lambda)^\top U_\lambda$, and its expected product with the transformed nuisance score is zero by this equation. Smoothness, nonsingular nuisance information and suitable initial coordinates provide the local ordinary-differential-equation construction. Simultaneous orthogonalization for several interest coordinates may face integrability constraints.
For normal location and variance $(\mu,\sigma^2)$, the cross information is zero: the location score is proportional to the centered observation and the variance score to its squared value minus its expectation, whose product has zero expectation by normal symmetry. This is a useful orthogonal parameterization. In general orthogonality reduces nuisance sensitivity and motivates adjusted profile constructions, but it implies neither exact estimator independence nor an exact nuisance-free conditional likelihood.
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