Understanding Products of Inertia

We covered this topic in Dynamics yesterday and the text does a less than average job on the topic. I am confused on the argument of symmetry. Does this mean if an item is symmetrical with respect to all planes the Inertia is 0 ( even when it is not bisected by the planes in the axis given. We looked at an example in class dealing with three pipes joined by 90 degree elbows. Two of which laid on the given x and the given y axis. The third however in the z direction laid under the x and y plane and was off of the given z axis. We used the parallel axis theorem which I understand. It's just that pipe in the z direction had a zero in the product of inertia. and I thought that it would have some sort of inertia since it was not centered on the point of rotation. Any help clearing this up would be greatly appreciated. Thanks!
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