While this is from the Hisorical Iris Society, it still explains just how tough it is to id an Iris. There are thousands and thousands of cultivars.

Here is a link that might be useful: identifying iris

Thank you grannymarsh!
I can see that identifying irises is not like identifying roses.
The website you recommended will give me tools for describing the iris, even though it is not likely I will be able to ID it.
Renee

It looks like a pretty rebloomer --- that alone (RE) will make your search easier.

Knowing what company you have gotten it from and what year may also help.

Carlos

Thanks Irisgal and Carlos. It's STILL BLOOMING.
Irisgal, I have enjoyed your posts, and Carlos, I think about you whenever I look at the clump of iris that has been in the front yard since my grandparents bought the place in 1960. It was an old clump then, and my mom brought it back to life. When I moved in in 1989, I found a few rhizomes under an oleander and now they are back in business. One plant gave me 28 blooms one year!
Renee

One possible match is Matrix. But of course the only way to be sure is to buy Matrix and any other potential matches and grow them side by side with your unknown iris in your own garden so that you can make comparisons of all plant and growth characteristics to see if they are identical.

Laurie

I found it!
It's called "Total Recall".
It's got two new bloom stalks.