  We consider the semilinear Lane-Emden problem
\begin{equation}\label{problemAbstract}\left\{ \begin{array}{lr} -\Delta u=
|u|^{p-1}u\qquad \mbox{ in }\Omega\\ u=0\qquad\qquad\qquad\mbox{ on }\partial
\Omega \end{array} \right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and
$\Omega$ is a smooth bounded symmetric domain of $\mathbb R^2$. We show that
for families $(u_p)$ of sign-changing symmetric solutions of
\eqref{problemAbstract} an upper bound on their Morse index implies
concentration of the positive and negative part, $u_p^\pm$, at the same point,
as $p\to+\infty$. Then an asymptotic analysis of $u_p^+$ and $u_p^-$ shows that
the asymptotic profile of $(u_p)$, as $p\to+\infty$, is that of a tower of two
different bubbles.
