We directly measured the three-dimensional dynamics of nematic disclination lines during the relaxation process from an electrically driven turbulent state, by using confocal microscopy and fluorescent dye accumulation on disclinations. We focused, in particular, on reconnections of disclinations. We showed the scaling law for in-plane reconnection, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection. Moreover, we revealed that reconnecting disclinations approach symmetrically, which is contrasted with asymmetric dynamics known for the two-dimensional counterpart. We discuss this symmetry restoring mechanism in terms of both topology and energy and expect it to have generality beyond liquid crystals.