Naturally the pivot allows the seat to be folded up when the seat is removed, as can be seen on the left.
The seat, meanwhile, is a simple plywood board (bottom right), with a strengthening middle section added to thicken it, and two end lips that enable it to lock onto the upper crosspieces. 
I thought about using a flexible leather seat, as seems to be the case in some illustrations, and may use one on a future fauldstool, but the solidity of this form of construction is an asset. This seat has a cushion made for it, though I tend to drape a sheepskin over the top for extra comfort. The seat is often evicted at night (without its soft furnishings) to make for more room in the pavilion, and in many weeks of soaking and dew, hasn't come to any harm so far.
A big part of Medieval re-enactment is trying to remind people of the tastes of our pre-Reformation Catholic ancestors, so the fauldstool got a major dose of undercoat and paintwork after its manufacture, to better reflect the opulence and decoration of the 13th century, and in order to hide the pine nature of the base materials. Being a mobile piece of furniture, it's taken a few knocks over the years, and now looks well-travelled and campaigned. I love it, and few who see it don't ask to have a go on it!