  The so-called Kalton-Peck space $Z_2$ is a twisted Hilbert space induced,
using complex interpolation, by $c_0$ or $\ell_p$ for any $1\leq p\neq
2<\infty$. Kalton and Peck developed a scheme of results for $Z_2$ showing that
it is a very rigid space. For example, every normalized basic sequence in $Z_2$
contains a subsequence which is equivalent to either the Hilbert copy $\ell_2$
or the Orlicz space $\ell_M$. Recently, new examples of twisted Hilbert spaces,
which are induced by asymptotic $\ell_p$-spaces, have appeared on the stage.
Thus, our aim is to extend the Kalton-Peck theory of $Z_2$ to twisted Hilbert
spaces $Z(X)$ induced by asymptotic $c_0$ or $\ell_p$-spaces $X$ for $1\leq
p<\infty$. One of the novelties is to use spreading models to gain information
on the isomorphic structure of the subspaces of a twisted Hilbert space. As a
sample of our results, the only spreading models of $Z(X)$ are $\ell_2$ and
$\ell_M$, whenever $X$ is as above and $p\neq 2$.
