  A Hankel operator $\mathbf{H}_\varphi$ on the Hardy space $H^2$ of the unit
circle with analytic symbol $\varphi$ has minimal norm if
$\|\mathbf{H}_\varphi\|=\|\varphi \|_2$ and maximal norm if
$\|\mathbf{H}_\varphi\| = \|\varphi\|_\infty$. The Hankel operator
$\mathbf{H}_\varphi$ has both minimal and maximal norm if and only if
$|\varphi|$ is constant almost everywhere on the unit circle or, equivalently,
if and only if $\varphi$ is a constant multiple of an inner function. We show
that if $\mathbf{H}_\varphi$ is norm-attaining and has maximal norm, then
$\mathbf{H}_\varphi$ has minimal norm. If $|\varphi|$ is continuous but not
constant, then $\mathbf{H}_\varphi$ has maximal norm if and only if the set at
which $|\varphi|=\|\varphi\|_{\infty}$ has nonempty intersection with the
spectrum of the inner factor of $\varphi$. We obtain further results
illustrating that the case of maximal norm is in general related to "irregular"
behavior of $\log |\varphi|$ or the argument of $\varphi$ near a "maximum
point" of $|\varphi|$. The role of certain positive functions coined apical
Helson--Szeg\H{o} weights is discussed in the former context.
