We suggest new strict constraints that the two-particle cumulant matrix should fulfill. The constraints are obtained from the decomposition of ⟨Ŝ(2)⟩, previously developed in our laboratory, and the vanishing number of electrons shared by two non-interacting fragments. The conditions impose stringent constraints into the cumulant structure without any need to perform an orbital optimization procedure thus carrying very small or no computational effort. These constraints are tested on the series of Piris natural orbital functionals (PNOF), which are among the most accurate ones available in the literature. Interestingly, even though all PNOF cumulants ensure correct overall ⟨Ŝ(2)⟩ values, none of them is consistent with the local spin structure of systems that dissociate more than one pair of electrons. A careful analysis of the local spin components reveals the most important missing contributions in the cumulant expression thus suggesting a means to improve PNOF5. The constraints provide an inexpensive tool for the construction and testing of cumulant structures that complement previously known conditions such as the N-representability or the square of the total spin angular momentum, ⟨Ŝ(2)⟩.