We obtain symmetry-extended counting rules for the mobility of general plate-bar frameworks in configurations with non-trivial point-group symmetry. Necessary conditions for isostaticity of a symmetric rod-bar framework in 3-space are derived. An example shows that establishing sufficient conditions will require significant further development. A symmetry-extended counting rule is established for rod-clamp frameworks: plate-bar frameworks are clamped in such a way as to remove relative translations within clamped pairs. Worked examples showing the utility of the symmetry approach in detecting mechanisms and states of self-stress include an application to linear pentapods where a singular configuration is detected by symmetry.