I am reading "Vector Mechanics for Engineers, Statics | Dynamics", 12th edition, by Beer, Johnston, Mazurek, Cornwell, Self. In section 18.3C Motion of an Axisymmetric Body Under No Force, the authors write (you can skip the text and focus only on the bold sentence that I confused about):
We can now analyze the motion of an axisymmetric body about its mass center under no force except its own weight. Examples of such motion are furnished by projectiles (if air resistance is neglected) and by satellites and space vehicles after the burnout of their launching rockets.
The sum of the moments of the external forces about the mass center G of the body is zero, so $\dot H_G$ = 0 . It follows that the angular momentum $H_G$ of the body about G is constant. Thus, the direction of $H_G$ is fixed in space and can be used to define the Z axis, or axis of precession (Fig. 18.20). Let us select a rotating system of axes Gxyz with the z axis along the axis of symmetry of the body, the x axis in the plane defined by the Z and z axes, and the y axis pointing away from you (Fig. 18.21). This gives us:
$$ H_x = −H_Gsin θ ; H_y = 0 ; H_z = H_G cos θ $$(18.46)
where θ represents the angle formed by the Z and z axes, and $H_G$ denotes the constant magnitude of the angular momentum of the body about G. Because the x, y, and z axes are principal axes of inertia for the body considered, we have
$$ H_x = I′ω_x ; H_y = I′ω_y ; H_z = Iω_z $$(18.47)
where I denotes the moment of inertia of the body about its axis of symmetry and I′ denotes its moment of inertia about a transverse axis through G. It follows from Eqs. (18.46) and (18.47) that
$$ \omega_x = \frac{H_Gsin\theta}{I'} ; \omega_y = 0 ; \omega_z = \frac{H_Gcos\theta}{I}$$(18.48)
The second of these relations shows that the angular velocity ω has no component along the y axis; that is, along an axis perpendicular to the Z-z plane. Thus, the angle θ formed by the Z and z axes remains constant and the body is in steady precession about the Z axis.
I thought principal axes give $I_{xy} = I_{yx} = I_{xz} = I_{zx} = I_{yz} = I_{zy} = 0$, but look at the figure 18.20: we only have $I_{xy} = I_{yx} = I_{yz} = I_{zy} = 0$ and I think $I_{xz} = I_{zx} \neq 0$ So why are the x, y, and z axes principal axes of inertia for the body considered ?