How does this multiple-curve-beam flexural pivot work? I found this in Handbook of Compliant Mechanisms (2013), page 162, or at the start of &quot;Chapter 11, Elements of Mechanisms,&quot; subsection 11.1.2 Revolute
I don't understand how it's supposed to go from 1 to 2 when b rotates around c.
The description reads:

This element is a rotational flexural pivot constructed by three
curved beams to achieve a large range of motion. Theoretically, this
element will rotate without axial-drift motion, because of the
symmetric arrangement about the axis.
(1) Rigid body a is fixed. Rigid body b rotates about c-axis.
(2) Deformed configuration
(3) Photo of the device.

It's unclear which part of the beam is attached to what. I can understand how one curved beam could switch its curvature (in general, like they do in  bi-stable latches), but I don't see how they could both at the end of of the rotation end up curved in a way that's opposite to how they started.
How could c2/c3 go from the configuration in 1 to the configuration in 2 ?
Or could they be two different iterations of the same idea ? I can see how (1) or (2) would resist rotation of c around b, and snap it back to its original position.  The text claims (2) is the deformed configuration though. I can also see how (1) with just c1, c3, c5 would deform to (1) with c0, c2, c4 if (b) was turning anti clockwise.
Also, what would be an ideal material to print this kind of compliant mechanism ?
The picture looks as if there is a printed back surface to which the central part is attached.  This can not be the case, since otherwise the curved members would not be able to flex.  Everything inside the outer ring must be detached from the back shell.
Like all couplers, especially flexible couplers, there is a limit to the amount of torque the coupler can transmit.  Within the elastic limits of the material this looks like a good design, and a good match for 3D printing.  The forces are along the layers, not across the layers.
This looks like a good design for a rotary coupler.  I'm resisting calling it a &quot;shaft&quot; coupler since neither side is equipped with any connection to a shaft.  One could modify the design to have a larger central hub with a shaft hole (and set screw(s)).  As it is, there is an implied method to hold the outer ring, and a fairly explicit three-pin adapter to drop into the open slots in the arms which connect to the center.
I would use ABS rather than PLA, although it depends on the stiffness you require and ability to sustein abuse.  PLA is stiller than ABS, while ABS, within the elastic limits, is more compliant.  I am not confident that either of these plastics would stand up to thousands of millions of flexures.
I would prefer to print this of TPU or Nylon.  Both of these are tougher than ABS and PLA.  They withstand greater flex with fewer problems with micro cracks and degradation.  I have printed another shaft coupler of TPU, and it was both compliant and still.
It isn't possible to really nail down a material without knowing the application.
As to your question about one example being a transform of the other, I don't thing they are.  The A and B drawings are similar in function, but are not stressed and unstressed versions of the same part.  Either will work as a coupler.