  A coupling between Earths rotation and orbital angular momentum (OAM), known
as the Sagnac effect, is observed in entangled neutrons produced using a spin
echo interferometer. After correction for instrument systematics the measured
coupling is within 5% of theory, with an uncertainty of 7.2%. The OAM in our
setup is transverse to the propagation direction and scales linearly with
wavelength (4 A - 12.75 A), hence the coupling can be varied, without
mechanically rotating the device. Therefore, the systematic error is lower than
in previous experiments. The detected transverse OAM of our beam corresponds to
4098 +- 295 hbar A-1, 5 orders of magnitude lower than in previous neutron
experiments, thereby demonstrating the feasibility of using the Sagnac effect
to definitively measure neutron OAM and paving the way towards observations of
the quantum Sagnac effect
