Orthogonal coefficient blocks in a centered normal linear model ($X_1^{\mathsf{T}}{X}_2=0$)

In a normal linear model whose predictor columns are orthogonal to the intercept, two coefficient blocks are orthogonal statistical parameters precisely when the cross-block Fisher information is zero, equivalently the displayed design cross-product vanishes. Then $X^{\mathsf{T}}X$ is block diagonal, so the corresponding least-squares estimators have zero covariance and, being jointly Gaussian, are independent. Their fitted subspaces are orthogonal and their extra sums of squares do not depend on which block is added first.