  We study the relationship of zonal gravity coefficients, J_2n, zonal winds,
and axial moment of inertia (MoI) by constructing models for the interiors of
giant planets. We employ the nonperturbative concentric Maclaurin spheroid
(CMS) method to construct both physical (realistic equation of state and
barotropes) and abstract (small number of constant-density spheroids) interior
models. We find that accurate gravity measurements of Jupiter's and Saturn's
J_2, J_4, and J_6 by Juno and Cassini spacecrafts do not uniquely determine the
MoI of either planet but do constrain it to better than 1%. Zonal winds (or
differential rotation, DR) then emerge as the leading source of uncertainty.
For Saturn, they are predicted to decrease the MoI by 0.4% because they reach a
depth of ~9000 km while on Jupiter, they appear to reach only ~3000 km. We thus
predict DR to affect Jupiter's MoI by only 0.01%, too small by one order of
magnitude to be detectable by the Juno spacecraft. We find winds primarily
affect the MoI indirectly via the gravity harmonic J_6 while direct
contributions are much smaller because the effects of pro- and retrograde winds
cancel. DR contributes +6% and -0.8% to Saturn's and Jupiter's J_6 value,
respectively. This changes the J_6 contribution that comes from the uniformly
rotating bulk of the planet that correlates most strongly with the predicted
MoI. With our physical models, we predict Jupiter's MoI to be 0.26393+-0.00001.
For Saturn, we predict 0.2181+-0.0002, assuming a rotation period of 10:33:34 h
that matches the observed polar radius.
