  We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on
${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0
< \alpha <\frac {4} {N-2}$, for every $\mu >0$, there exist infinitely many
sign-changing, self-similar solutions to the Cauchy problem with initial value
$u_0 (x)= \mu |x|^{-\frac {2} {\alpha }}$. The construction is based on the
analysis of the related inverted profile equation. In particular, we construct
(sign-changing) self-similar solutions for positive initial values for which it
is known that there does not exist any local, nonnegative solution.
