  In this paper, we investigate a kind of special quintom model, which is made
of a quintessence field $\phi_1$ and a phantom field $\phi_2$, and the
potential function has the form of $V(\phi_1^2-\phi_2^2)$. This kind of quintom
fields can be separated into two kinds: the hessence model, which has the state
of $\phi_1^2>\phi_2^2$, and the hantom model with the state
$\phi_1^2<\phi_2^2$. We discuss the evolution of these models in the
$\omega$-$\omega'$plane ($\omega$ is the state equation of the dark energy, and
$\omega'$ is its time derivative in unites of Hubble time), and find that
according to $\omega>-1$ or $<-1$, and the potential of the quintom being
climbed up or rolled down, the $\omega$-$\omega'$ plane can be divided into
four parts. The late time attractor solution, if existing, is always
quintessence-like or $\Lambda$-like for hessence field, so the Big Rip doesn't
exist. But for hantom field, its late time attractor solution can be
phantom-like or $\Lambda$-like, and sometimes, the Big Rip is unavoidable. Then
we consider two special cases: one is the hessence field with an exponential
potential, and the other is with a power law potential. We investigate their
evolution in the $\omega$-$\omega'$ plane. We also develop a theoretical method
of constructing the hessence potential function directly from the effective
equation of state function $\omega(z)$. We apply our method to five kinds of
parametrizations of equation of state parameter, where $\omega$ crossing -1 can
exist, and find they all can be realized. At last, we discuss the evolution of
the perturbations of the quintom field, and find the perturbations of the
quintom $\delta_Q$ and the metric $\Phi$ are all finite even if at the state of
$\omega=-1$ and $\omega'\neq0$.
