Coordinate cannot be symmetric. Only a molecule can.

Except of very primitive cases there is no point to look on internal coordinates, various visualisation software can identify point group of your molecule with given thresholds. For example Gaussview or Chemcraft.

Energy has nothing to do to symmetry.

Thank you for your answer. Nevertheless, I am not interested in the point group of my molecule. What I would like to really know is whether if I take an arbitrary axis around which a part of the molecule is rotated with arbitrary angles +delta and -delta, then I get two structures of the same energy. Nothing else is necessary for me. I do not need programs, but a method that is able to answer this question.

Answer is still strongly related to previous one.

So you have 2 parts of molecule, connected with a bond.

If initial geometry has plane of symmetry in which lies the bond, around which you perform rotation than this rotation in both directions will be exactly the same.

In case if parts of the molecule don't have some strong noncovalent interaction your rotation can be almost symmetric also when given criteria is not fullfilled.

Few examples:

Toluene H3C-C6H5, in case if in starting geometry there is dihedral H-C-C-C (first 2 atoms of methyl group, last 2 of phenyl ring) equal to 0 or 90 degrees (so mollecule has plane of symmetry) - clockwise and counterclockwise rotation of methyl group will give EXACTLY same energy (up to numerical noise).

H3C-C6H4-OOH (peroxo group in para position, rest of molecule has mentioned symmetry plane). Molecule is chiral, so theoretically rotation of C-C bond is NOT symmetric. However it will be VERY close to symmetry.

H3C-C6H4-OOH (peroxo group in orto position, dihedral C1-C-O-O where C1 is connected to methyl group < 90 degrees) - effect will be much more pronounced.

C6H5-C6H5 in minimum geometry (torsion angle C-C-C-C 44.4°) - rotation around central bond will be significantly asymmetric with respect to starting point but symmetric in respect to angle 0 or 90 degrees.

Proline (nonzwitterionic form) - rotation of carboxyl group will be noticably asymmetric independently on starting geometry

So for getting theoretical answer you need to identify planes of symmetry of your molecule.

For practically useful answer you need to identify noncovalent close contacts of atoms in different fragments (that are mainly not listed as internal coordinates) and estimate their effect.

P.S. more exact formulation of criteria would be:

If both fragments have plane of symmetry that includes rotation bond, then rotation will be symmetric in respect to geometry in which these planes completely overlap.

Thank you again. My molecules are weakly bound van der Waals dimers (A.B) and, of course, I try to study their coordinates in those regions where the intermolecular interactions can be enforced.

Let pA and pB are two (arbitrary) symmetry planes of monomers A and B, respectively. If I understand correcly what you wrote, the rotation around an arbitrary axis R is symmetric only in the case that R is contained both by pA and pB, and pA and pB totally overlap. Then, for a given R, I have to find a plane of A, pA, that contains R, and search for a plane of B, pB, that coincides with pA. Is it a good "algorithm" to inspect whether R is symmetric?