One of the classical and difficult problems in the theory of planar differential systems is to classify their centers. Here we classify the global phase portraits in the Poincare disc of the class continuous piecewise differential systems separated by one straight line and formed by two ℤ2 -equivariant cubic Hamiltonian systems with nilpotent bi-centers at (±1,0). The main tools for proving our results are the Poincare compactification, the index theory, and the theory of sign lists for determining the exact number of real roots or negative real roots of a real polynomial in one variable.