In the infrared limit, a nearly anti-de Sitter spacetime in two dimensions (AdS_{2}) perturbed by a weak double trace deformation and a two-site (q>2)-body Sachdev-Ye-Kitaev (SYK) model with N Majoranas and a weak 2r-body intersite coupling share the same near-conformal dynamics described by a traversable wormhole. We exploit this relation to propose a symmetry classification of traversable wormholes depending on N, q, and r, with q>2r, and confirm it by a level statistics analysis using exact diagonalization techniques. Intriguingly, a time-reversed state never results in a new state, so only six universality classes occur-A, AI, BDI, CI, C, and D-and different symmetry sectors of the model may belong to distinct universality classes.