In general, rearranging the legs of a Stewart-Gough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather unexpected way. Nevertheless, some leg rearrangements have been recently found to leave singularities invariant but, unfortunately, these rearrangements are only valid for Stewart-Gough platforms containing rigid components. In this work, the authors go a step further presenting singularity-invariant leg rearrangements that can be applied to any Stewart-Gough platform whose base and platform attachments are coplanar. The practical consequences of the presented theoretical results are illustrated with several examples including well-known architectures.