Ferroelectric Ising chain magnet Ca$_3$Co$_{2-x}$Mn$_x$O$_6$ ($xsimeq$0.96) was studied in magnetic fields up to 33 T. Magnetization and neutron scattering measurements reveal successive metamagnetic transitions from the zero-field $uparrow uparrow downarrow downarrow$ spin configuration to the $uparrow uparrow uparrow downarrow$ state with a broad magnetization plateau, and then to the $uparrow uparrow uparrow uparrow$ state. The absence of hysteresis in these plateaus reveals an intriguing coupling between the intra-chain state and the three-dimensional geometrically frustrated magnetic system. Inversion symmetry, broken in the $uparrow uparrow downarrow downarrow$ state, is restored in the $uparrow uparrow uparrow downarrow$ state, leading to the complete suppression of the electric polarization driven by symmetric superexchange.
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