Hi Paul
The centre cane is intriguing, but I always start from the 'It's a squished cane' position, and then try to find an alternative explanation. If only that were a more common approach we would have far less fantasy in eBay descriptions!
Your centre cane could well have nothing more than a distorted 4 lobe cross in the centre - it has a high degree of symmetry. I wrote an article on how Walsh Walsh and Arculus, when making a simple 4 lobe cross cane for Old English weights, could end up with the well known "7/6" cane, "window" canes, and various letter and number canes! It is in the 2006 Annual Bulletin of the PCA, and called 'Alphabet Soup: The Origins of Old English "7/6" Canes.
I would not rule out your centre cane being something made specially - but my money is presently on 'squished cane'. If the person setting the canes out saw an attractive design in a cane, albeit made by accident, they might well use it. I'm sure Clichy often did that with bits of cane that looked to have a 'C' in them.
Alan