  An accelerating boundary (mirror) acts as a horizon and black hole analog,
radiating energy with some particle spectrum. We demonstrate that a M\"obius
transformation on the null coordinate advanced time mirror trajectory uniquely
keeps invariant not only the energy flux but the particle spectrum. We clarify
how the geometric entanglement entropy is also invariant. The transform allows
generation of families of dynamically distinct trajectories, including
$\mathcal{PT}$-symmetric ones, mapping from the eternally thermal mirror to the
de Sitter horizon, and different boundary motions corresponding to Kerr or
Schwarzschild black holes.
