  The Lorenz-96 model is widely used as a test model for various applications,
such as data assimilation methods. This symmetric model has the forcing
$F\in\mathbb{R}$ and the dimension $n\in\mathbb{N}$ as parameters and is
$\mathbb{Z}_n$ equivariant. In this paper, we unravel its dynamics for $F<0$
using equivariant bifurcation theory. Symmetry gives rise to invariant
subspaces, that play an important role in this model. We exploit them in order
to generalise results from a low dimension to all multiples of that dimension.
We discuss symmetry for periodic orbits as well.
  Our analysis leads to proofs of the existence of pitchfork bifurcations for
$F<0$ in specific dimensions $n$: In all even dimensions, the equilibrium
$(F,\ldots,F)$ exhibits a supercritical pitchfork bifurcation. In dimensions
$n=4k$, $k\in\mathbb{N}$, a second supercritical pitchfork bifurcation occurs
simultaneously for both equilibria originating from the previous one.
  Furthermore, numerical observations reveal that in dimension $n=2^qp$, where
$q\in\mathbb{N}\cup\{0\}$ and $p$ is odd, there is a finite cascade of exactly
$q$ subsequent pitchfork bifurcations, whose bifurcation values are independent
of $n$. This structure is discussed and interpreted in light of the symmetries
of the model.
