  The paper concerns several theoretical aspects of oriented supersingular
$\ell$-isogeny volcanoes and their relationship to closed walks in the
supersingular $\ell$-isogeny graph. Our main result is a bijection between the
rims of the union of all oriented supersingular $\ell$-isogeny volcanoes over
$\overline{\mathbb{F}}_p$ (up to conjugation of the orientations), and isogeny
cycles (non-backtracking closed walks which are not powers of smaller walks) of
the supersingular $\ell$-isogeny graph over $\overline{\mathbb{F}}_p$. The
exact proof and statement of this bijection are made more intricate by special
behaviours arising from extra automorphisms and the ramification of $p$ in
certain quadratic orders. We use the bijection to count isogeny cycles of given
length in the supersingular $\ell$-isogeny graph exactly as a sum of class
numbers of these orders, and also give an explicit upper bound by estimating
the class numbers.
