Inside-out polygonal dissections A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new polygon $P'$.
Say that a polygon $P$ has an inside-out dissection (my terminology)
if $P'$ is congruent to $P$, and the perimeter of $P$ becomes interior to $P'$,
and so the perimeter of $P'$ is composed of internal cuts of the dissection of $P$.
I believe every polygon $P$ has an inside-out dissection because
(1) $P$ may be triangulated, and (2) every triangle has an inside-out dissection:

 
 
 


One may ask many questions concerning this concept. Here I will confine myself to three:


Q0. Has this notion been explored before, and if so, under what name?
Q1. Is there an inside-out dissection of a generic triangle using fewer than $9$ pieces?
Q2. There is a "$+$" inside-out dissection of any rectangle into $4$ pieces.
  What is the minimal inside-out dissection of a generic trapezoid?

 A: Your triangle dissection into $9$ pieces can be modified: .
As masterfully noted below, The dissection into $16$ congruent triangles similar to the big triangle can be fused into four pieces. Swap the two triangles and rotate each parallelogram $180^{\circ}.$

A: Do you allow mirror reflection to count as congruent? Because my paper Hinged Kite Mirror Dissection (arXiv preprint, 2001) dissects and reassembles an arbitrary polygon to become its mirror image; all exterior edges of the original polygon become interior edges of the dissection. So this is sort of an answer to Q0. Additionally (unlike your triangulation argument) the dissection is hinged. But the paper doesn't explicitly talk about your inside-out property.
A: Riffing off the tiling comment to another answer, imagine a square penny packing of circles,
and then translate the tiling so that a circle is in the center of four other circles.  Now replace
each of the five circles with a translate of a sufficiently convex (but not necessarily regular!) polygon,
and you will for some polygons get a five piece inside out dissection.
A: Here is another view of the @AaronMeyerowitz / @TheMaskedAvenger 4-piece inside-out
dissection of a triangle:

 
 
 


