rotational kinematics - Can the direction of angular momentum and angular velocity differ?

While studying rotational mechanics, I came across a section where it mentioned that angular momentum may not necessarily be parallel to angular velocity. My thoughts were as follows:

Angular momentum ($L$) has the relation $L=I\omega$ where $\omega$ is angular velocity and $I$ is the moment of inertia, so following this relation, it seems they should be in the same direction. Why are they not?

Answer

Consider a thin rectangular block with width $w$, height $h$ resting along the xy plane as shown below.

The mass of the block is $m$. The mass moment of inertia (tensor) of the block about point A is

$${\bf I}_A = m \begin{vmatrix} \frac{h^2}{3} & -\frac{w h}{4} & 0 \\ -\frac{w h}{4} & \frac{w^2}{3} & 0 \\ 0 & 0 & \frac{w^2+h^2}{3} \end{vmatrix}$$

This was derived from the definition (as seen on https://physics.stackexchange.com/a/244969/392)

If this block is rotating along the x axis with a rotational velocity

$$\boldsymbol{\omega} = \begin{pmatrix} \Omega \\ 0 \\ 0 \end{pmatrix}$$ then the angular momentum about point A is

$${\bf L}_A = m \Omega\,\begin{pmatrix} \frac{h^2}{3} \\ -\frac{w h}{4} \\ 0 \end{pmatrix}$$

As you can see, there is a component of angular momentum in the y direction. The angular momentum vector forms an angle $\psi = -\tan^{-1} \left( \frac{3 w}{4 h} \right)$

In the figure below you see the direction of angular momentum, and the circle about which the center of mass is going to orbit due to precession.