I know Byzantine Catholics will bless themselves with the right side first in order to honour Christ who is seated at the right hand of the Father. They will also take their thumb, index finger and middle finger and press them together to honour the trinity, while the last two fingers are tucked away in their palms n order to honour Christ’s two natures. They will alsomake the cross three times in order to honour the trinity.
I’m unsure though how this explanation came about. If I had to guess, I would say that the explanation above came about as pious explanation of why Christians make the cross in the manner they do.
I’ve heard it said that blessing from right to left is the more ancient practice, but I’ve also come across material that suggests that both practices are almost the same age. Otherwise there is the popular explaination that it changed from right to left, to left to right in the West because the laity would mirror the priest who was facing them. I personally think that its a weak argument for various reasons.
At the end of the day, I think the historical record is a bit cloudy on the issue. If you look hard enough you’ll find a wide variety of explanations out there.