  We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c
l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 &
\textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega
\subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$,
$|u|^{q-2}u$ will be identified with $\sgn(u)$. We establish a variational
principle for least energy nodal solutions, and we investigate their
qualitative properties. In particular, we show that they satisfy a unique
continuation property (their zero set is Lebesgue-negligible). Moreover, if
$\Omega$ is radial, then least energy nodal solutions are foliated Schwarz
symmetric, and they are nonradial in case $\Omega$ is a ball. The case $q=1$
requires special treatment since the formally associated energy functional is
not differentiable, and many arguments have to be adjusted.
