Meaning: impossible to reconcile; "irreconcilable differences"
The cross is the conflict of two hostile lines, of irreconcilable direction.
However, I put the bag in the back of my mind, which was fast becoming stored with anomalous and apparently irreconcilable facts, and followed Warner to the house.
That appeared to make the whole future hideous. To break with Osmond once would be to break for ever; any open acknowledgement of irreconcilable needs would be an admission that their whole attempt had proved a failure.