  Oscillator networks display intricate synchronization patterns. Determining
their stability typically requires incorporating the symmetries of the network
coupling. Going beyond analyses that appeal only to a network's automorphism
group, we explore synchronization patterns that emerge from the phase-shift
invariance of the dynamical equations and symmetries in the nodes. We show that
these nonstructural symmetries simplify stability calculations. We analyze a
ring-network of phase-amplitude oscillators that exhibits a "decoupled" state
in which physically-coupled nodes appear to act independently due to emergent
cancellations in the equations of dynamical evolution. We establish that this
state can be linearly stable for a ring of phase-amplitude oscillators, but not
for a ring of phase-only oscillators that otherwise require explicit
long-range, nonpairwise, or nonphase coupling. In short, amplitude-phase
interactions are key to stable synchronization at a distance.
