From the equation:
The x and the y component are the same, and the ‘mixing’ components xy and yx are zero, so there is no difference between direction perpendicular on the axis, so the system is axis symmetrical.
From the figure:
By turning the dump-bell around the z-axis, the distance between all the charges stays the same, this is the quantity that determines the interaction and thus the energy. So turning the dump-bell around the z-axis, will not affect the energy, so the system is axis symmetricaL.