Spectral (sub)manifolds are nonlinear, dynamics-invariant surfaces that are locally tangent to an anchor invariant subspace attached to a hyperbolic equilibrium solution of the governing equations. In systems exhibiting dynamically-dominant slow time scales, these surfaces allow for the construction of reduced-order models (ROMs) capable of providing an accurate description of the asymptotic dynamics of the flow. Specifically, a ROM is obtained by constraining the state of the full-order system to evolve on the aforementioned manifold according to polynomial intrinsic dynamics computed using a graph-style parameterization. Following this approach, dynamically-consistent intrinsic (or latent--space) coordinates can be easily identified by projecting the state onto the manifold along linear fibers that are orthogonal to a properly defined left (or adjoint) spectral subspace. We apply this formulation to a canonical, yet notoriously challenging, incompressible axisymmetric jet flow driven by a time-periodic inflow boundary condition. For this configuration, the flow exhibits a Floquet-stable time-periodic solution characterized by a street of unpaired vortices. However, due to its underlying non-normality, the flow strongly amplifies external disturbances, leading to large-amplitude transient growth and strong nonlinear behavior featuring vortex pairing and merging. We demonstrate that ROMs evolving on spectral manifolds outperform models of comparable dimensions evolving on linear subspaces, and are capable of capturing the vortex pairing phenomenon both in the linear (i.e., onset) and nonlinear (i.e., fully-developed) regimes.