  We report on progress made in the construction of higher-derivative
superinvariants for type-II theories in ten dimensions. The string amplitude
calculations required for this analysis exhibit interesting features which have
received little attention in the literature so far. We discuss two examples
from a forthcoming publication: the construction of the (H_{NS})^2 R^3 terms
and the fermionic completion of the \epsilon\epsilon R^4 terms. We show that a
correct answer requires very careful treatment of the chiral splitting theorem,
implies unexpected new relations between fermionic correlators, and most
interestingly, necessitates the use of worldsheet gravitino zero modes in the
string vertex operators. In addition, we discuss the relation of our results to
the predictions of the linear scalar superfield of the type-IIB theory and find
(and explain) an interesting discrepancy.
