@C_R

L1 := 1.6; L2 := 1.5; L5 := 1.35;
f1 := (x7-g1)^2+(x8-g2)^2+(x9-g3)^2-L1^2;
f2 := (x10-g1)^2+(x11-g2)^2+(x12-g3)^2-L1^2;
f3 := (x1-CD1)^2+(x2-CD2)^2+(x3-CD3)^2-L1^2;
f4 := (x4-CD1)^2+(x5-CD2)^2+(x6-CD3)^2-L1^2;
f5 := (x4-x1)^2+(x5-x2)^2+(x6-x3)^2-L2^2;
f6 := (x7-x10)^2+(x8-x11)^2+(x9-x12)^2-L2^2;
f7 := ((x1+x4)*(1/2)-(x7+x10)*(1/2))^2+((x2+x5)*(1/2)-(x8+x11)*(1/2))^2+((x3+x6)*(1/2)-(x9+x12)*(1/2))^2-L5^2;
f8 := (x10-x7)*(g2-x8)-(x11-x8)*(g1-x7);
f12 := x4*(CD2-x2)-x5*(CD1-x1);
f9 := (x1-x4)*(x7-x10)+(x2-x5)*(x8-x11)+(x3-x6)*(x9-x12);
f10 := ((x7+x10)*(1/2)-(x1+x4)*(1/2))*(x4-x1)+((x8+x11)*(1/2)-(x2+x5)*(1/2))*(x5-x2)+((x9+x12)*(1/2)-(x3+x6)*(1/2))*(x6-x3);
f11 := ((x7+x10)*(1/2)-(x1+x4)*(1/2))*(x10-x7)+((x8+x11)*(1/2)-(x2+x5)*(1/2))*(x11-x8)+((x9+x12)*(1/2)-(x3+x6)*(1/2))*(x12-x9);
T := Isolate([f1, f2, f3, f4, f5, f6, f6, f7, f8, f9, f10, f11, f12], [x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12])

L1 := 1.6 is the length of the levers; L2 := 1.5 is the width of the horns; L5 := 1.35 is the distance between the midpoints of the horns. And some of these parameters are printed (after calculations) in one of the given programs (in particular, L5).

The system of equations itself gives 32 solutions, but it seems that there are only 4 fundamental provisions themselves. In other words, the system of equations does not contain free variables, and, accordingly, we do not have degrees of freedom of mechanism. Of course, we are talking about a specific mathematical model.But discarding f7, we formally get one degree of freedom, and with it a slight deformation of the "yellow" lever (in our case, L5)

To be honest, I made this mechanism out of curiosity. I was interested in testing exactly this approach. And this is where my knowledge ends.

If we talk about inverse kinematics, then I do not quite understand what "switching" is. But, I think that we can always replace the right construction (rhombus) with the left one. It is necessary to add equations corresponding to a rhombus to the mathematical model of the left device, and then the left device will work like a rhombus. That is, we virtually remove unnecessary degrees of freedom. This, of course, if I understand your question correctly.

I often look at the forum, but did not see your message, because the "flag" for me was not red. For this it is necessary. to have addressed to me. How do you now have a red flag.

Thanks again for your interest.