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