I know that both are supposed to mean "to place" but are they interchangeable? I know that 摆 can be used in the following manner:
So could you also say something like
and have it be a valid sentence?
lay, layout, place, set
 [v] lay; put; place; arrange
 [v] display; assume; put on
 [v] state clearly; talk about
 [v] wave; sway; swing; wag
 [n] oscillating device; pendulum
 let go; release; set free
 let out; dismiss; take leave
 put; place; set; lay
 let off; set off; send out
 distribute; hand out; lend for interest
 send into exile; exile; banish
 (of flowers) blossom; come into bloom
 enlarge; lengthen; let out/down
 graze; pasture; put out to feed
The topic is the verb "to place". Both 擺 and 放 can be the verb for "to place", but 擺 also implies the object is placed in a particular position (arranged)
You can say 擺下陣勢 ( lay down a formation) but not 放下陣勢, because a formation has to be arranged.
桌子上摆着三本书。( more likely be neatly arranged) 桌子上放着三本书 ( either randomly or neatly positioned)
Both are valid but with subtle differences. 摆 is more like "put for display", while 放 is just "put".