  We study a specific correlation in spherical harmonic multipole domain for
cosmic microwave background (CMB) analysis. This group of correlation between
\Delta l=4n, n=1,2... is caused by symmetric signal in the Galactic coordinate
system. An estimator targeting such correlation therefore helps remove the
localized bright point-like sources in the Galactic plane and the strong
diffused component down to the CMB level. We use 3 toy models to illustrate the
significance of these correlations and apply this estimator on some derived CMB
maps with foreground residuals. In addition, we show that our proposed
estimator significantly damp the phase correlations caused by Galactic
foregrounds. This investigation provides the understanding of mode correlations
caused by Galactic foregrounds, which is useful for paving the way for
foreground cleaning methods for the CMB.
