= Solution
Decompose $\mathbf u=\mathbf u_p+u_\phi\mathbf e_\phi$ and $\mathbf B=\mathbf B_p+B_\phi\mathbf e_\phi$. In a steady <axisymmetric vector field>, $\nabla\times\mathbf E=0$ and $\mathbf E=-\mathbf u\times\mathbf B$. The radial and vertical components of its <curl> are
$$
-\partial_zE_\phi=0,\qquad R^{-1}\partial_R(RE_\phi)=0.
$$
Therefore $RE_\phi=C$ is a constant. Regularity at the symmetry axis sets $C=0$; equivalently, one imposes no toroidal loop <electromotive force>, $\oint\mathbf E\cdot d\mathbf l=2\pi C=0$. Under this usual condition,
$$
0=E_\phi=-(\mathbf u_p\times\mathbf B_p)_\phi,
$$
which implies
$$
\boxed{\mathbf u_p=\alpha\mathbf B_p}
$$
where $\mathbf B_p\ne0$. This is the <regular-axis alignment of steady poloidal ideal flow>. It allows nonzero toroidal components as well.
The axis or zero-loop-voltage condition is necessary for the unqualified conclusion. On an annular domain, $\mathbf B=(B_0/R)\mathbf e_R$ and $\mathbf u=U_0\mathbf e_z$ give $\mathbf E=-(U_0B_0/R)\mathbf e_\phi$ with zero curl, yet the poloidal vectors are perpendicular. This can satisfy the remaining steady equations as well: choose constant <mass density> $\rho_0$, $\Phi=\pi G\rho_0R^2$ and $p=p_0-\pi G\rho_0^2R^2>0$ on the annulus. The field has zero <curl>, radial <pressure> balances gravity, the flow advects no equilibrium quantity, and $\nabla^2\Phi=4\pi G\rho_0$. It carries a nonzero loop voltage. Steady axisymmetry alone on an annulus therefore does not exclude such cross-field flow.
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