A pure fermionic state with a fixed particle number is said to be correlated if it deviates from a Slater determinant. In the present work we show that this notion can be refined, classifying fermionic states relative to $k$-${\rm \textit{body}}$ correlations. We capture such correlations by a family of measures $ω_k$, which we call twisted purities. Twisted purity is an explicit function of the $k$-fermion reduced density matrix, insensitive to global single-particle transformations. Vanishing of $ω_k$ for a given $k$ generalizes so-called Plücker relations on the state amplitudes and puts the state in a class ${\cal G}_k$. Sets ${\cal G}_k$ are nested in $k$, ranging from Slater determinants for $k = 1$ up to the full $n$-fermion Hilbert space for $k = n + 1$. We find various physically relevant states inside and close to ${\cal G}_{k=O(1)}$, including truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in ${\cal G}_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.