The cycles of Euclidean geometry are: Points, lines, circles.
The cycles of hyperbolic geometry are: Points, lines, circles, hypercycles, horocycles.
There is a commonality between these: These are all orbits of some point under a connected Lie subgroup of the congruence group. In Euclidean geometry, the congruence group is $E(2)$, and in hyperbolic geometry it's $PGL(2,\mathbb R)$.
The problem is that there is some ambiguity in what a hyperbolic hypercycle is: Under one definition, it has two connected components instead of one. But then it's not the orbit of a point under a connected Lie subgroup. If this definition of a hyperbolic hypercycle gets adopted, then what then does the word "cycle" mean in general?
Further thoughts
What happens if you use a Cayley-Klein model? In a CK geometry (which both Euclidean and hyperbolic geometries are), a cycle could perhaps be defined as a quadric closed under a connected Lie subgroup of the congruence group. Then a hyperbolic hypercycle has two connected components instead of one, as desired.