A friend has 8 of these growing in his garden. He was told recently that it's Epipactis helleborine, but the leaves don't match. The flowers do appear to be orchid-like.
Looks like it. See this link (link removed) [Edited by wcutler May 24, 2020]: that link to missouriplants was not working. Here is one to eFloras Plants of Missouri: Epipactis helleborine in Flora of Missouri @ efloras.org
The flowers look the same, but the leaves on the Epipactis helleborine look like orchid leaves, and those on the mystery plant look almost corn-like.
Some recent research over here has found that Epipactis is made up of numerous microspecies; a number of new local endemic species have been described. If this applies in North America too, there's always even a chance it is an undescribed new one.
Produces both broad and skinny leaves. http://www.mobot.org/plantscience/ResBot/Flor/WNY-Niag/epiphell.jpg A UK site calls it broad-leaf helleborine, leaf shape might be one of the characters used to split it up into microspecies, ones producing skinny leaves only perhaps being separated out.
Thank you all. If you click on the thumbnail of my picture to enlarge it, you will see that the leaf axils are very grass-like, not like the leaves of epipactus. That's what is so puzzling.
