We present a simulation study of how properties of symmetric diblock copolymers depend on the invariant degree of polymerization N¯, focusing on the vicinity of the order-disorder transition (ODT). Results from several coarse-grained simulation models are combined to cover a range of N¯ ≃ 200-8000 that includes most of the experimentally relevant range. Results are presented for the free energy per chain, the value of πeN at the ODT, the latent heat of transition, the layer spacing, the composition profile, and compression modulus in the ordered phase. Universality (i.e., model independence) is demonstrated by showing that equivalent results for all properties are obtained from corresponding thermodynamic states of different simulation models. Corresponding states of symmetric copolymers are states with equal values of the parameters πeN and N¯, where πe is an effective Flory-Huggins interaction parameter and N is a degree of polymerization. The underlying universality becomes apparent, however, only if data are analyzed using an adequate estimate of πe, which we obtain by fitting the structure factor in the disordered state to recent theoretical predictions. The results show that behavior near the ODT exhibits a different character at moderate and high values of N¯, with a crossover near N¯ ≃ 104. Within the range N¯ ≲ 104 studied here, the ordered and disordered phases near the ODT both contain strongly segregated domains of nearly pure A and B, in contrast to the assumption of weak segregation underlying the Fredrickson-Helfand (FH) theory. In this regime, the FH theory is inaccurate and substantially underestimates the value of πeN at the ODT. Results for the highest values of N¯ studied here agree reasonably well with FH predictions, suggesting that the theory may be accurate for N¯ ≳ 104. Self-consistent field theory (SCFT) grossly underestimates (πeN)ODT for modest N¯ because it cannot describe strong correlations in the disordered phase. SCFT is found, however, to yield accurate predictions for several properties of the ordered lamellar phase.