  The spontaneous breaking of chiral symmetry implies the presence of a
short-distance scale in the QCD vacuum, which phenomenologically may be
associated with the "size" of the constituent quark, rho ~ 0.3 fm. We discuss
the role of this scale in the matrix elements of the twist-4 and 3 quark-gluon
operators determining the leading power (1/Q^2-) corrections to the moments of
the nucleon spin structure functions. We argue that the flavor-nonsinglet
twist-4 matrix element, f_2^{u - d}, has a sizable negative value of the order
rho^{-2}, due to the presence of sea quarks with virtualities ~ rho^{-2} in the
proton wave function. The twist-3 matrix element, d_2, is not related to the
scale rho^{-2}. Our arguments support the results of previous calculations of
the matrix elements in the instanton vacuum model. We show that this
qualitative picture is in agreement with the phenomenological higher-twist
correction extracted from an NLO QCD fit to the world data on g_1^p and g_1^n,
which include recent data from the Jefferson Lab Hall A and COMPASS
experiments. We comment on the implications of the short-distance scale rho for
quark-hadron duality and the x-dependence of higher-twist contributions.
