Is moment of inertia additive? If so, why doesn't adding two halves of solid box work?

I’m doing a project where I want to calculate the moment of inertia for some objects. I’ve broken the objects down into simple objects like cubes, spheres, cylinders, etc — things I know the moment of inertia tensor for. My plan was to take these smaller parts and "glue them together." Since I know the inertia tensors for these simpler shapes and I thought moment of inertia was additive, I thought I could sum them and get the moment of inertia for the composite object. However, the following thought experiment has convinced me I’m mistaken.

Suppose I have a solid box (aka cuboid) with height $h$, width $w$, length $ell$, and mass $m$. Now, suppose I cut this solid box in two, so the height is now $tfrac{1}{2}h$ for each half box and similarly the mass for each is $tfrac{1}{2}m$. The moment of inertia tensor for a solid box is

$$mathbf{I}_{cube} = begin{bmatrix}
frac{1}{12} m (h^2 + d^2) & 0 & 0
0& frac{1}{12} m (d^2 + w^2) & 0
0& 0& frac{1}{12} m (w^2 + h^2)
end{bmatrix}
$$

So for each half box, denoted by an index $iin lbrace 1,2rbrace$, the tensor should be

$$mathbf{I}_i = begin{bmatrix}
frac{1}{12} left(frac{m}{2}right) left( left(frac{h}{2}right)^2 + d^2right) & 0 & 0
0& frac{1}{12} left(frac{m}{2}right) (d^2 + w^2) & 0
0& 0& frac{1}{12} left(frac{m}{2}right) left(w^2 + left(frac{h}{2}right)^2right)
end{bmatrix}
$$

So now suppose I meld the two halves back together, thus restoring my initial cube. If moment of inertia is additive, then I should be able to sum $mathbf{I}_1$ and $mathbf{I}_2$ and get the original tensor of the cube. However, you can see this doesn’t happen.

$$begin{align}
mathbf{I}_1 + mathbf{I}_2 &=
2 begin{bmatrix}
frac{1}{12} left(frac{m}{2}right) left( left(frac{h}{2}right)^2 + d^2right) & 0 & 0
0& frac{1}{12} left(frac{m}{2}right) (d^2 + w^2) & 0
0& 0& frac{1}{12} left(frac{m}{2}right) left(w^2 + left(frac{h}{2}right)^2right)
end{bmatrix}&=
begin{bmatrix}
frac{1}{12} mleft( left(frac{h}{2}right)^2 + d^2right) & 0 & 0
0& frac{1}{12} m (d^2 + w^2) & 0
0& 0& frac{1}{12} m left(w^2 + left(frac{h}{2}right)^2right)
end{bmatrix}
&ne mathbf{I}_{cube}
end{align}
$$

Why is this the case? In what sense is moment of inertia additive?