  We prove the existence of nonradial solutions for the H\'enon equation in the
ball with any given number of nodal zones, for arbitrary values of the exponent
$\alpha$. For sign-changing solutions, the case $\alpha=0$ -- Lane-Emden
equation -- is included. The obtained solutions form global continua which
branch off from the curve of radial solutions $p\mapsto u_p$, and the number of
branching points increases with both the number of nodal zones and the exponent
$\alpha$. The proof technique relies on the index of fixed points in cones and
provides information on the symmetry properties of the bifurcating solutions
and the possible intersection and/or overlapping between different branches,
thus allowing to separate them at least in some cases.
