  We revisit the symmetry structure of integrable PDEs, looking at the specific
example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending
on a parameter, which we call generating symmetries. We explain that since
these are nonlocal symmetries, their commutator algebra is not uniquely
determined, and we present three possibilities for the algebra. In the first
version, 3 of the 4 symmetries commute; this shows that it is possible to add
further (nonlocal) commuting flows to the standard KdV hierarchy. The second
version of the commutator algebra is consistent with Laurent expansions of the
symmetries, giving rise to an infinite dimensional algebra of hidden symmetries
of KdV. The third version is consistent with asymptotic expansions for large
values of the parameter, giving rise to the standard commuting symmetries of
KdV, the infinite hierarchy of "additional symmetries", and their traditionally
accepted commutator algebra (though this also suffers from some ambiguity as
the additional symmetries are nonlocal). We explain how the 3 symmetries that
commute in the first version of the algebra can all be regarded as
infinitesimal double B\"acklund transformations. The 4 generating symmetries
incorporate all known symmetries of the KdV equation, but also exhibit some
remarkable novel structure, arising from their nonlocality. We believe this
structure to be shared by other integrable PDEs.
