  We investigate spin-refined partition functions in AdS/CFT using Euclidean
gravitational path integrals. We construct phase diagrams for $Z_X = \text{Tr}
\big( e^{-\beta H} X \big)$ in various dimensions and for different choices of
discrete isometry $X$, discovering rich structures at finite temperature. When
$X$ is a reflection, $Z_X$ counts the difference between the number of even-
and odd-spin microstates. The high-temperature regime is universally dominated
by $\mathcal{CRT}$-twisted black holes in any dimension, and in odd spacetime
dimensions we examine whether complex rotating black hole solutions could
contribute to spin-refined observables. We also analyze the microcanonical
ensemble. There the leading contribution almost always come from rotating black
holes, showing that the two ensembles are not equivalent.
