  We derive equations of motion for Green's functions of the multi-orbital
Anderson impurity model by differentiating symmetrically with respect to all
time arguments. The resulting equations relate the one- and two-particle
Green's function to correlators of up to six particles at four times. As an
application we consider continuous-time quantum Monte Carlo simulations in the
hybridization expansion, which hitherto suffered from notoriously high noise
levels at large Matsubara frequencies. Employing the derived symmetric improved
estimators overcomes this problem.
