Does anyone know the origins of the Duke of Wellington's Regt. cap badge and beret?

I had heard that the officers wore square badges as at Waterloo they had formed squares long with the men whilst the other ranks had a triangular backing to the cap badge as they formed "triangles" when there were too few men to support the square. I was also told that other ranks were forbidden from sewing the backing on to the beret for some misdemeanour at some point.

However, I have looked through some regimental literature and found nothing to confirm this. Is it just a rumour? Does anyone know the story behind it?
I was only a cadet and enquiries at battalion were normally met with, "I have heard that" but no verification as to how true it was. It always seemed odd to me that the backing was a different shape and it made me think there may be some truth in it but never gt a definitive answer

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