Cutting rectangles and squares into non-rectilinear tiles

A rectilinear polygon is one with all angles either 90 or 270 degrees. This post continues the overflow posts:
this and this

Question: Can a square (or even some rectangle) by tiled with an odd number N of mutually congruent polygons that are non-rectilinear? The tiles could be non-convex.

Note 1: If N is even, a square can be easily cut into say 4 mutually congruent and non-orthogonal non-convex pieces by 4 identical polylines that connect the center to the 4 vertices.

Note 2: this discussion is on (mostly) tiling rectangles with odd number of rectilinear tiles.

At the end is a question by Victor Protsak: "Do you know if there is a tiling of a rectangle into an odd number of congruent convex n-gons for n>=5?" That is pretty much what we asked above, except for relaxing convexity that is!