Backward Letters as another group?
 

One of the Alpha Phi Omega Fratenrity chapters about 50 years ago founded Omega Phi Alpha Sorority which has gone onto be its own national organization.

As far as I can tell, this makes Alpha Phi Omega relatively unique in that a mirror image of its letters written horizontally is *another* organization. (and yes, Alpha Phi Omega did accidentally publish a backward picture in our magazine one issue) (I don't think Omega Tau Alpha or Omega Chi Alpha exist)

The only other organization that I can think of where writing the letters in the other direction is another national group is Kappa Sigma/Sigma Kappa, but both of those letters aren't symmetrical, so a Kappa Sigma sweatshirt in a mirror doesn't look like Sigma Kappa...

Am I missing any other mirror/backward pairs?

It looks like there's a Phi Alpha honor society which would mirror Alpha Phi, right? This is fun, I'm going to think of more (if they exist)

I believe there's a Phi Beta Pi med school honorary.

Google says there's a Pi Delta Alpha Mass Communications fraternity at Texas Tech.

Both have Wikipedia pages.
Phi Alpha is an honor society at Illinois College that started in the 1840s and still exists today.
Phi Beta Pi medical school honorary was originally called Pi Beta Phi, but changed its name due to the NPC group.

While not exactly a mirror image, I saw an event shirt that had the following.

SK
KS
(then the event name)

So if you read it by rows, it read (top row) Sigma Kappa, (bottom row) Kappa Sigma. If you read it by columns, it also read (column 1) Sigma Kappa, (column 2) Kappa Sigma.

Maybe it was unintentional to have this kind of/sort of symmetry, but I thought it was kind of neat. (Yes, I used the word, “neat”.)

According to our HQ social media one our upcoming colonizations was able to borrow Kappa Sigma letters to promote Sigma Kappa. :)

You have APO/OPA covered.

Alpha Phi Alpha is the same both ways.

Gamma Xi Phi backward is Phi Xi Gamma and there doesn't seem to be one.

There is no IIOA, but we call AOII husbands/boyfriends/significant others IIOAs.

Does it have an "Acronym"? Pals Of Actives?

Nope, just Pi-O-As. It started back in the 60s, and some chapters even made up "initiation rituals" to commemorate new marriages. They stopped talking about IIOAs for many years, but recently it has seen a resurgence with IHQ even making shirts for sale in the AOII Emporium.