  We compare order $R^4$ terms in the 10-dimensional effective actions of
SO(32) heterotic and type I superstrings from the point of view of duality
between the two theories. Some of these terms do not receive higher-loop
corrections being related by supersymmetry to `anomaly-cancelling' terms which
depend on the antisymmetric 2-tensor. At the same time, the consistency of
duality relation implies that the `tree-level' $R^4$ super-invariant (the one
which has $\zeta(3)$-coefficient in the sphere part of the action) should
appear also at higher orders of loop expansion, i.e. should be multiplied by a
non-trivial function of the dilaton.
