  The chiral expansion of the $\pi\pi$ amplitude to the order of two loops was
expressed in terms of six independent parameters in a previous paper: four of
these are shown here to satisfy sum rules. Their derivation, where crossing
symmetry plays a key role, is explained. Their convergence properties are
studied and their practical evaluation, in terms of the available data on
$\pi\pi$ phase shifts above 0.5 GeV, is discussed. Below 0.5 GeV, the chiral
amplitude itself is employed, such that the parameters are determined in a
self-consistent way. Some care is devoted to the estimate of the errors.
