  If a $Spin(7)$ manifold $N^8$ admits a free $S^1$ action preserving the
fundamental $4$-form then the quotient space $M^7$ is naturally endowed with a
$G_2$-structure. We derive equations relating the intrinsic torsion of the
$Spin(7)$-structure to that of the $G_2$-structure together with the additional
data of a Higgs field and the curvature of the $S^1$-bundle; this can be
interpreted as a Gibbons-Hawking-type ansatz for $Spin(7)$-structures. We focus
on the three $Spin(7)$ torsion classes: torsion-free, locally conformally
parallel and balanced. In particular we show that if $N$ is a $Spin(7)$
manifold then $M$ cannot have holonomy contained in $G_2$ unless $N$ is in fact
a Calabi-Yau $4$-fold and $M$ is the product of a Calabi-Yau $3$-fold and an
interval. We also derive a new formula for the Ricci curvature of
$Spin(7)$-structures in terms of the torsion forms. We then describe this
$S^1$-quotient construction in detail for the Bryant-Salamon $Spin(7)$ metric
on the spinor bundle of $S^4$ and for the flat metric on $\mathbb{R}^8$.
