The cover polynomial, as introduced by \textit{F. R. K. Chung} and \textit{R. L. Graham} [J. Comb. Theory, Ser. B 65, No. 2, 273-290 (1995; Zbl 0839.05045)], generalizes the factorial rook polynomial to two variables, by also encoding information about the partial permutations which correspond to rook placements. It is a well-known fact due to \textit{J. R. Goldman, J. T. Joichi} and \textit{D. E. White} [Proc. Am. Math. Soc. 52, 485-492 (1975; Zbl 0312.05002)] that the factorial rook polynomial of any Ferrers board, or, more generally, of any permutation of the columns of a Ferrers board (called skyline board by the author), factors completely into linear factors. The purpose of the article under review is to examine how much of this can be extended to the cover polynomial. The author shows that there is a straightforward extension to increasing Ferrers boards. The cover polynomial of a decreasing Ferrers board factors as well, although the analysis in this case is much more difficult. For arbitrary skyline boards there is no complete factorization into linear factors anymore. Still, through a very tricky analysis, the author establishes a (partial) factorization theorem in this case as well. The paper closes with factorization theorems for the cover polynomial of column-permuted staircase boards.