The title pretty much sums up the question. I can't see any reason why the orientation of an oil cooler should matter, but if there's a reason, I'm betting that someone here will weigh in on it.
The installation instructions don't indicate a preferred orientation, but all the diagrams show the cooler in a horizontal orientation, with the fittings pointing in the "up" direction.
For my application, it looks like the cooler will fit better in a vertical orientation, with the fittings pointing in a "sideways" direction.
Steve.