  A fundamental theorem of Laman characterises when a bar-joint framework
realised generically in the Euclidean plane admits a non-trivial continuous
deformation of its vertices. This has recently been extended in two ways.
Firstly to frameworks that are symmetric with respect to some point group but
are otherwise generic, and secondly to frameworks in Euclidean 3-space that are
constrained to lie on 2-dimensional algebraic varieties.
  We combine these two settings and consider the rigidity of symmetric
frameworks realised on such surfaces. By extending the orbit matrix techniques
of [32, 12], we prove necessary conditions for a framework to be
symmetry-forced rigid (i.e., to have no non-trivial symmetry-preserving motion)
for any group and any surface. In the cases when the surface is a sphere, a
cylinder or a cone we use Henneberg-type inductive constructions on
group-labeled quotient graphs to prove that these conditions are also
sufficient for a number of symmetry groups, including rotation, reflection,
inversion and dihedral symmetry. For the remaining groups - as well as for
other types of surfaces - we provide some observations and conjectures.
