A closer look at the solutions 

Using our result we define a function valid[a_,b_,c_]:=8 (a^2b^2)^2 (a^2+b^2)5 c^6+11 (a^2b^2)^2 c^2+2 (a^2+b^2) c^4 which is >= 0 if the kaleidocycle works with the hinges at edge c and <0 else. Without loss of generality we can set c=1 in the following region plot of valid(a,b,c) (hinges at edge c), valid(b,c,a) (hinges at edge a) and valid(c,a,b) (hinges at edge b). In red regions we have values >=0. npoints=400;


We now search for a few extremal points, which all lie on the boundary of the red region, and there we have valid[a,b,c]=0.


There is a class of tetrahedra which is called Heronian tetrahedra. Here all edge lengths, face areas and the volume have integer values. The subclass of Heronian disphenoids plays an important role in the theory of Heronian tetrahedra. This motivates a similar question: which integer lengths disphenoids can serve as the basis for a kaleidocycle? The following code computes the number a(n) of disphenoids which can serve as kaleidocycle which have integer edge lengths <= n. value[a_,b_,c_]:=8 (a^2b^2)^2 (a^2+b^2)5 c^6+11 (a^2b^2)^2 c^2+2 (a^2+b^2) c^4 {0,1,3,6,10,15,21,28,36,44,54,66,80,96,113,132,153,176,200,225} (OEIS A338334 ) There are for example three possible disphenoids with edge lenghts <=3: {2,2,1}, {3,3,1} and {3,3,2}. The other triangles do not work: {1,1,1}, {2,2,2} and {3,3,3} are acute but blocking occurs with the corresponding kaleidocycles. {3,2,2} is not acute and hence does not allow a disphenoid. And {2,1,1} , (3,1,1) and {3,2,1 }are no triangles. If we request scalene triangles for the faces of the disphenoid which have integer edge lengths <= n we get a[n_]:=Module[{a,b,c,t=0},Do[If[value[a,b,c]>=0,t++],{c,n},{b,c+1,n},{a,b+1,n}];t]; {0,0,0,0,0,0,0,0,0,0,1,3,6,10,14,19,25,32,40,48,57,68,82,97,113,131,151,173,196,220} (OEIS A338335 ) The only scalene possible triangle with edge lenghts <= 11 is {11,10,8}. If we are looking for integer triangles for the faces of the disphenoid which have perimeter n we get a[n_]:=Module[{a,b,c,t=0},Do[If[a=nbc;a>=b&&value[a,b,c]>=0,t++],{c,Quotient[n,3]},{b,c,nc}];t]; {0,0,0,0,1,0,1,1,1,1,2,1,2,2,2,2,3,2,3,3,3,3,4,3,4,4,4,4,5,4,6,5,6,6,7,6,7,7,7,8,8,8,9,9,9,10,10,9,11,10} The smalles possible perimeter is 5 with the triangle {2,2,1}. 

Are there integer length triangles such that with the corresponding kaleidocycle neighbouring faces touch 4 times during a cycle? This would mean equality in the relation between a, b and c: I did not find an answer to this question. 

References 1. Doris Schattschneider and Wallace Walker, M.C. Escher Kaleidocycles, 1977. ISBN 0906212286 

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