  We use modular invariance and crossing symmetry of conformal field theory to
reveal approximate reflection symmetries in the spectral decompositions of the
partition function in two dimensions in the limit of large central charge and
of the four-point function in any dimension in the limit of large scaling
dimensions $\Delta_0$ of external operators. We use these symmetries to
motivate universal upper bounds on the spectrum and the operator product
expansion coefficients, which we then derive by independent techniques. Some of
the bounds for four-point functions are valid for finite $\Delta_0$ as well as
for large $\Delta_0$. We discuss a similar symmetry in a large spacetime
dimension limit. Finally, we comment on the analogue of the Cardy formula and
sparse light spectrum condition for the four-point function.
