  We clarify the global geometry of two 1-parameter families of cohomogeneity
one Spin(7) holonomy metrics with generic orbit the Aloff--Wallach space
$N(1,-1) \cong \mathrm{SU}(3)/\mathrm{U}(1)$ and singular orbits $S^5$ and
$\mathbb{C}P^2$, which at short distance were shown to exist by Reidegeld. The
two families fit into the geography of previously known families of
cohomogeneity one metrics with exceptional holonomy and provide a Spin(7)
analogue of the well-known conifold transition in the setting of Calabi--Yau
3-folds. Furthermore, we discover that there is another transition to families
of Spin(7) holonomy metrics which have a similar asymptotic behaviour on one
end, but are singular on the other end. We obtain our results by relating the
Spin(7)-equations to a simple dynamical system on a 3-dimensional cube.
