Thanks to a very helpful comment left by Mommy Bee in which her father gave some extremely helpful clues as to the variety of my mystery bearded iris, I believe I have finally tracked down a positive identification!
I think it's a Wabash!
The photo and description at White Flower Farm really cinched it. Here's what it says:
"A look of tailored elegance is the gift this heirloom Iris brings to the bed or border. 'Wabash' was first introduced in 1936 and became a favorite Bearded Iris of the 1940s and '50s. When the sweetly fragrant blossoms open in June, it's easy to understand why. The crisp white of the standards (the smaller, erect petals) poses a striking contrast to the reddish purple of the pendulous falls, which are finished round the edge with a fine, pale piping."
So I've got an heirloom iris and I didn't even know it.