Can something cause both breaks and cross-links in DNA?
Cisplatin forms DNA-protein crosslinks (so DNA-protein-DNA by extension) in vitro [1] and a review [2] says that cisplatin leads to DSBs (although I can't access the primary source at the moment). But you know, with the exception of ionizing radiation, mutagens usually don't directly cause DSBs. It's caused by repair processes or things like stalling at replication forks because of the mutagen.

[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1874601/

[2] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4047912/#!po=11.9919