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Abstract Fix an integer$$d\ge 2$$ $d\ge 2$. The parameters$$c_0\in \overline{\mathbb {Q}}$$ $c_0\in \overline{Q}$for which the unicritical polynomial$$f_{d,c}(z)=z^d+c\in \mathbb {C}[z]$$ $f_{d,c}\left(z\right)=z^d+c\in C\left[z\right]$has finite postcritical orbit, also known asMisiurewiczparameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work from a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we callp-special, which may be of independent number theoretic interest.