3D kinematic geometry problem motivated by chemistry It is well known that six carbon atoms can form a ring called cyclohexane. Since the angle between bonds is $\cos^{-1}\left(\frac{-1}{3}\right)\approx 109^\circ$, the ring is not a planar hexagon.  There is a flexible configuration called the boat and a rigid one called the chair.  
I am interested in the $3D$ geometry. The chemistry is modeled by rods attached at nodes so that the angle at which two rods meet is fixed but dihedral angles are free to change.  
Question: In such a structure, where all the rods are equal and all the angles are equal, for which angles does there exist a flexible structure? Furthermore, classify all such structures, flexible or rigid.
I know that for $120^\circ$, the planar hexagon, there are no flexible structures.
Conversations with several chemists have yielded no information.
 A: There is a trivial answer to the question as posed.  Since you are allowing spherical ball-socket joints between the rods, in any stable configuration, any of the rods can rotate freely around its longitudinal axis. I'm not a chemist, I'm a 3D geometry problem-solver, so I don't know what the implications are.
I'm guessing you are looking for 3D kinematic freedom that amounts to more than the trivial case I've cited?  You're looking for flexibility where at least one joint is moving in relation to the others?
That, too, spawns a simplistic solution.  If you consider a "V" of two jointed rods, as a rigid unit, it can rotate axially around the straight-line axis running between the outer endpoint joints, only impeded by collision avoidance with the other elements of the structure. 
I'm guessing here, but you want flexibility in the structure where the joint angles (measured as 2D angles) are flexing? 
That, too, is trivial to achieve.  Pick 2 joints opposite each other hexagonally,
freeze all the other joint angles, and allowing just these two freedom.  Each half-hex can be visualized as a rigid arc, and allow one to rotate axially about the straight-line axis bridging between their arc endpoints.  This will flex the joint angles where the half-hex structures are freely jointed.  Collisions between the arcs would delimit the range of rotation.
Maybe I'm not understanding the constraints, but I see no reason why a planar hexagonal structure cannot be flexed as follows: 
Attach equal length pull-cords to joints 1,3,5, tie the other ends together, from where you can tug on all 3 joints equally at the same time.  Using 3 more cords, bind joints 2,4,6 so they can be pulled on as one unit in the opposite direction.  Now stretch the pull cords, and the hex will contract into a "VVV" (triple-U), until all the space between the joints has collapsed.  You'll have joints 1,3,5 tightly packed at one end, and joints 2,4,6 packed together at the other end.
The process is reversible by compressing, rather than extending, the hex structure. By symmetry, all 6 angles change in unison.
And, you can combine the above 4 manipulations into a dizzying explosion of 3D possibilities, and it would be hard looking at any to figure out how you got it to bend that way.  The only one that forces unequal joint angles is the half-hex arcs.
Beyond that, could I suggest you build a manipulative with the properties you specify, and learn by playing with it? 
A: As a chemist, I just want to highlight that in actual chemical systems a number of things are different: the carbons also have hydrogens attached to them, and their rotations are limited by repulsions between the various atoms long before there is a collision. 
In the case of a 120 degrees chemical hexagon, i.e. benzene, there is no flexibility because the carbons have double bonds between them and they cannot freely rotate about C-C axis because of electronic structure considerations.
As a pure 3D geometry problem, the answer by pbierre seems quite complete to me.
