They also collaborated with the Pennsylvania Railroad at some point, considering that in Newark, they shared trackage, and at some point, the PRR owned some of their cars as some sported PRR logos.
The only reason the Port Authority took control of the PATH was that the PA wanted to build the WTC and as part of an agreement between NY & NJ, they took the failing railroad over and kept it running and did upgrades in order to be allowed to build the original World Trade Center which occupied the site of the H&M Terminal.