I would say that all organizations contain some spirituality in that there is bowing to the kamiza ...
Is the rei necessarily spiritual at all? My interpretation is that the baseline is acknowledgment of, and respect for, what has come before, and that is how I explain shomen-rei to my class. You can add whatever sentiment you like on top of that.
I like Kanshu Sunadomari's statement that aikido practice itself can be thought of as a spiritual exercise. Again I don't believe that it is necessarily so for everyone, but I think I understand what Sunadomari Sensei means.