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There are a lot of questions in my (first year) lecture notes where the instructor uses moment of inertia tensor to perform calculations involving angular momentum. And he directly makes a statement such as Ixy =0 ... I have no idea how he reached that conclusion just by observing and neither does he clarify one bit.

Is there any symmetry consideration to be taken into account? I know that say, if an object lies in the xz plane its product of inertia Ixy and Iyz will be zero but how to know that for the image file I've uploaded?enter image description here

Here the only term that remains in calculation is Izx while all of Ixx , Iyy , Izz , Iyz , Ixy turn out to be zero. HOW??

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For any 3D body it is always possible to find 3 mutually orthogonal axes to which the off diagonal terms are zero.

In your example that is not done and so the off diagonal terms are not necessarily zero.

$I_{\rm zx} = \int zx\,dm$ where you are adding all the contributions of masses $dm$ at position (x,y,z).

enter image description here

With the axes as chosen in your example (left hand diagram) that integral has a net negative value because the contributions from negative regions $A$ and $C$ are larger than that from the positive regions $B$ and $D$.

If the axes had been chosen along the principal axes the off diagonal value are all zero.
In the right hand diagram positive regions $B$ and $D$ contribute an equal amount to the integral as negative regions $A$ and $C$.

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