We build on the recent characterisation of congruences on the infinite twisted partition monoids P n Φ $\mathcal {P}_{n}^\Phi$ and their finite d $d$ -twisted homomorphic images P n , d Φ $\mathcal {P}_{n,d}^\Phi$ , and investigate their algebraic and order-theoretic properties. We prove that each congruence of P n Φ $\mathcal {P}_{n}^\Phi$ is (finitely) generated by at most ⌈ 5 n 2 ⌉ $\lceil \frac{5n}{2}\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice Cong ( P n Φ ) $\text{\sf Cong}(\mathcal {P}_{n}^\Phi )$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite anti-chains. By way of contrast, the lattice Cong ( P n , d Φ ) $\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )$ is modular but still not distributive for d > 0 $d>0$ , while Cong ( P n , 0 Φ ) $\text{\sf Cong}(\mathcal {P}_{n,0}^\Phi )$ is distributive. We also calculate the number of congruences of P n , d Φ $\mathcal {P}_{n,d}^\Phi$ , showing that the array ( | Cong ( P n , d Φ ) | ) n , d ⩾ 0 $(|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|)_{n,d\geqslant 0}$ has a rational generating function, and that for a fixed n $n$ or d $d$ , | Cong ( P n , d Φ ) | $|\text{\sf Cong}(\mathcal {P}_{n,d}^\Phi )|$ is a polynomial in d $d$ or n ⩾ 4 $n\geqslant 4$ , respectively.