Hi Linda! I used to train there, so I think we have the same base in the terminology. I think this is key:
Peter A Goldsbury
Gyaku te dori simply applies the same concept to the hand grab. ...
However, I have never come across this term in Japan.
I think in the US, maybe it is within the Iwama style, there is a terminology that is maybe "weird." I won't say "wrong" because we use it successfully (once you get the hang of it) but, we see here that Dr. Goldsbury doesn't think it a normal usage of the terms.
Here's how I understand it:
Katatedori means single hand grab, and if you go to lots of different styles' dojos, you might see it as cross OR same side grab. I think that is true. Michael Hackett has shown that the names vary.
You guys are using the term "katate dori" to specifically refer to the kind of single-hand grab that is same SIDES of the 2-person pair. Like, uke's left hand holding nage's right wrist in words sounds opposite, but looking at the pair on the mat, it is the same side of the pair, say the west side. whereas "gyakute dori" as you are using it crosses over.
If you think of the ai hanmi you are referring to as "normal" fighting arrangement, where the bodies fit together like a puzzle, then that's why the other one is "gyaku," it is the reverse of the comfortable (ai hanmi).
It probably doesn't sound normal to a Japanese speaker, but I think this is how these terms came to be in your style.