  The Operator Product Expansion provides expressions for the $n^{th}$ moments
of $g_1(x)$ and $g_2(x)$ in terms of hadronic matrix elements of local
operators for $n =$ odd integer. In some cases these matrix elements are
expected to be small leading to approximate sum rules for the {\em odd\/}
moments of $g_{1,2}(x)$. We have shown how, working in a field-theoretic
framework, one can derive expressions for the {\em even\/} moments of the {\em
valence\/} parts of $g_{1,2}(x)$. These expressions cannot be written as matrix
elements of {\em local\/} operators and do not coincide with the analytic
continuation to $n=$ even integer of the OPE results.
  Just as for the OPE one can in some cases argue that the hadronic matrix
elements should be small, leading to approximate sum rules for the moments of
the valence parts of $g_{1,2}(x)$. But, most importantly, for the case $n=2$ we
have proved rigorously that the hadronic matrix element vanishes, yielding the
exact ELT sum rule \int^1_0 dx\, x\left[g^V_1(x)+2g^V_2(x)\right]=0.
  We have argued that the convergence properties of this sum rule are good and
have discussed how it can be used to get information about $g_2(x)$ and as a
further test of QCD.
