On a symmetry relating gravity with antigravity

I investigate the impact of a "would be" fundamental symmetry of the laws of nature under the interchange of gravity and antigravity, on the understanding of negative energies in general relativity. For this purpose a toy model that is based on Einstein-Hilbert gravity with two minimally coupled self-interacting scalar fields is explored, where the second (exotic) scalar field with negative energy density may be regarded, alternatively, as an antigravitating field with positive energy. Spontaneous breakdown of reflection symmetry is then considered in order to discuss the implications the proposed "would be" fundamental symmetry might have for the vanishing of the cosmological constant. A possible connection of the gravity-antigravity symmetry with the so called quintom field is also explored.