  Miquel dynamics is a discrete time dynamics for circle patterns. Previous
work shows that the circle centers satisfy the dSKP equation on the octahedral
lattice $A_3$ and give rise to a real-valued Y-system / cluster structure. In
this article we show that half the intersection points satisfy the dSKP
equation as well, and we introduce two new real-valued Y-systems for Miquel
dynamics that involve only the intersection points. Therefore, the new
Y-systems are M\"obius invariant. We also show that the circle centers and
intersection points combined satisfy the dSKP equation on the 4-dimensional
octahedral lattice $A_4$. In addition, we present two more complex-valued
Y-systems for Miquel dynamics, which are real-valued in and only in the case of
integrable circle patterns. We also show that in the case of harmonic
embeddings, the old and the new Y-patterns coincide.
