I think those are what are called 'ever blooming', Greg. If I'm recalling correctly, plants that do this are missing a gene or inhibitor and will produce a bloom stalk when the fan reaches maturity and is actively growing no matter the time of year.
@Barbalee rebloom and repeat bloom are completely new stalks, so cutting the stalks wouldn't have any effect on whether you got repeat bloom. Those rebloom events occur on different rhizomes anyway.
After saying that, just recently a hybridizer had a stalk put out new blooms from old spent sockets. I think it might have been Tom Johnson, but I'm not sure. It was on a Facebook forum and I expect it to be causing some talk and discussion among hybridizers. It certainly will if the plant reproduces and the phenomenon repeats on in subsequent growing seasons.