Symmetry of an interacting and a non-interacting state of a quantum system

I was studying about perturbation theory when I read about the conditions to the state of a interacting system be orthogonal to the state of the non-interacting system.
That is, consider that our quantum system has a Hamiltonian: $$H = H_{0} + V$$ And let $$\Psi_{0}$$ be an eigenstate of $$H$$ and $$\phi_{0}$$ an eigenstate of $$H_{0}$$ (both being the respective ground states). So, I read that the product $$\langle \phi_{0} | \Psi_{0} \rangle$$ only will be different from zero if the states $$\Psi_{0}$$ and $$\phi_{0}$$ have the same symmetry. I don't know what it means. What would be a symmetry of a state? And how can I determine it?

In perturbation theory, $$|\psi_0\rangle$$ is derived from the known $$|\phi_0\rangle$$. Let's say that $$|\phi_0\rangle$$ is symmetric(or antisymmetric), then the $$|\psi_0\rangle$$ must also be symmetric(or antisymmetric) or the $$|\psi_0\rangle$$ will not be a correct state representation of $$H$$.