How to justify symmetry in a torus for the calculation of $\mathbf H$? The typical texbook example of finding $\mathbf H$ in a torus filled with a material with magnetic permitivity $\mu_0$ (of course we don't need this, that's to find $\mathbf B$ later) always starts like this:

"The symmetry of the problem suggests that the field lines of $\mathbf H$ inside the torus are circles centered in the torus' axis and the module depends only on the distance $r$ from the axis" and then they just take $\mathbf H$ out of the integral along a circle in Ampere's law to find $\mathbf H$.  

I find this argument is not detailed enough to convince me.
How can I justify the other components of $\mathbf H$ are zero (radial and vertical if using cylindrical coordinates)?
 A: The Biot-Savart law $$ \vec H = \frac{1}{4 \pi}\int_C \frac {d\vec l \wedge \vec r'}{\vert \vec r' \vert^3} $$ tells us that the contribution to $\vec H$ of a current element $d \vec l$ is perpendicular to  $d \vec l$  and the vector linking the point of measurement of $\vec H$ to the current element $d\vec l$. A toroidal coil can be approximated by a succession of circular current loops. The field measured in the torus in the plane of the current loop (winding) must be perpendicular to the plane of the winding or parallel to the axis of the torus. If the core has a high $\mu$, the boundary condition for the magnetic induction, i.e. the continuity of the normal component of $\vec B$ across a boundary, will make sure that the field lines stay axially oriented and do not leak out of the torus.
A: You know that $\nabla \cdot \vec{B} = 0$, so in the case of a homogeneous core, $\nabla \cdot \vec{H}=0$ and no field lines can begin or end. That rules out any field component travelling away from the axis in the $R$ direction, which leaves only a $z$ or $\phi$ component. But we know that the field produced by a current element is always at right angles to that current element, so that rules out a $\phi$ component. 
Then, because the divergence is zero, then $H_z$ does not depend on $z$. 
Finally, in the current-free region of the toroid, the curl of the H-field is also zero. This means that $H_z$ also cannot depend on $R$ or $\phi$. 
Note: In my answer I am taking the cylindrical $z$-axis to run along the axis of the coil, the $\phi$ direction to run azimuthally around the coil axis and the $R$ direction to be radially away from the coil axis. In this coordinate system the current is always in the $\phi$ direction and the H-field runs in the $z$ direction only.
There is possible confusion here if you think that the $z$ direction is "vertical" with an axis at the centre of the torus and the $\phi$ direction is azimuthally around that axis. In those coordinates the H-field is only in the $\phi$ direction but, it is difficult to specify the direction of the current, hence my choice of coordinates.
