  In our previous two papers, we studied (positive) 3D gadgets in origami
extrusions which create a top face parallel to the ambient paper and two side
faces sharing a ridge with two simple outgoing pleats. Then a natural problem
comes up whether it is possible to construct a `negative' 3D gadget from any
positive one having the same net without changing the outgoing pleats, that is,
to sink the top and two side faces of any positive 3D gadget to the reverse
side without changing the outgoing pleats. Of course, simply sinking the faces
causes a tear of the paper, and thus we have to modify the crease pattern.
There are two known constructions of negative 3D gadgets before ours, but they
do not solve this problem because their outgoing pleats are different from
positive ones. In the present paper we give an affirmative solution to the
above problem. For this purpose, we present three constructions of negative 3D
gadgets with a supporting triangle on the back side, which are based on our
previous ones of positive 3D gadgets. The first two are an extension of those
presented in our previous paper, and the third is new. We prove that our first
and third constructions solve the problem. Our solutions enable us to deal with
positive and negative 3D gadgets on the same basis, so that we can construct
from an origami extrusion constructed with 3D gadgets its negative using the
same pleats if there are no interferences among the 3D gadgets. We also treat
repetition/division of negative 3D gadgets under certain conditions, which
reduces their interferences with others.
