  We study the structure of the backward shift invariant and nearly invariant
subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is
not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which
contain the polynomials as a dense subspace (in particular, in the radial case)
all nontrivial backward shift invariant subspaces are of the form
$\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials
of degree at most $n$. In general, the structure of the nearly invariant
subspaces is more complicated. In the case of spaces of slow growth (up to zero
exponential type) we establish an analogue of de Branges' Ordering Theorem. We
then construct examples which show that the result fails for general Fock-type
spaces of larger growth.
