CO₂-forced surface warming in general circulation models (GCMs) is initially polar-amplified in the Arctic but not Antarctic—a largely hemispherically antisymmetric signal. Nevertheless, we show in CESM1 and eleven LongRunMIP GCMs that the hemispherically symmetric component of global-mean-normalized, zonal-mean warming (T∗sym) under 4×CO₂ changes weakly or becomes modestly more polar-amplified from the first decade to near-equilibrium. Conversely, the antisymmetric warming component (T∗asym) weakens with time in all models, modestly in some including FAMOUS but effectively vanishing in others including CESM1. We explore mechanisms underlying the robust T∗sym behavior with a diffusive moist energy balance model (MEBM), which given radiative feedback parameter (λ) and ocean heat uptake (𝒪) fields diagnosed from CESM1 adequately reproduces the CESM1 T∗sym and T∗asym fields. In further MEBM simulations perturbing λ and 𝒪, T∗sym is sensitive to their symmetric components only, and more to that of λ. A three-box, two-timescale model fitted to FAMOUS and CESM1 reveals a curiously short Antarctic fast-response timescale in FAMOUS. In additional CESM1 simulations spanning a broader range of forcings, T∗sym changes modestly across 2-16×CO₂, and T∗sym in a Pliocene-like simulation is more polar-amplified but likewise approximately time-invariant. Determining the real-world relevance of these behaviors—which imply that a surprising amount of information about near-equilibrium polar amplification emerges within decades—merits further study.