Is this so? I cannot think of a counter-example and it is too general a statement to prove.
A counterexample: two human hands, identical except that on is a left hand and the other is a right hand.
There are many more: a cube with two cylindrical protrusions, has the same area and volume no matter how you move the protusions around.
Are there smooth manifolds (excepting mirroring)?
Basically you cut a shape in two parts and glue theme on another one.
Generalising your construction: construct a shape with complementary protrusions (sort of a hermaphroditic shape), cut another shape along the protrusion plane and fit the two parts in the respective protrusions. It seems to me that there will always be an edge.
Why does there have to be an edge? We can join the two shapes together with a smooth fillet.
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