DNA-DNA cross-linking with formaldehyde?
The fact that there is no inter-strand cross-linking between different double strands might be just because the cross-linker cannot bridge the distance between amines of different bases on different double strands.

The formaldehyde based linking of nucleobases has been described by Chaw et al. (1980) where formaldehyde bridged a gap of app. 3 angstroms (2x 1.3 A for N-C bond).

Even if you take the most condensed form of DNA as in nucleosomes (see X-ray model of nucleosome with bound DNA) you still have at least 20 angstroms between the bases of different double strands.
At the same time there are only 3 angstroms between the nucleobases of the same double strand, agreeing well with the length of a methyl-bridge.