Isn't that the purpose of the priviledge and immunities clause? States are prohibited from denying a person the priviledges and immunities that the person is entitled to as a citizen of the United States.
Well, sure, but these privileges and immunities, as they apply to citizens specifically not to persons in general, have a very specific meaning. They're not the same as rights, which are much harder to nail down. For further details, see post 28.