Energetically low-lying structural isomers of the much-studied thiolate-protected gold cluster Au25(SR)18- are discovered from extensive (80 ns) molecular dynamics (MD) simulations using the reactive molecular force field ReaxFF and confirmed by density functional theory (DFT). A particularly interesting isomer is found, which is topologically connected to the known crystal structure by a low-barrier collective rotation of the icosahedral Au13 core. The isomerization takes place without breaking of any Au-S bonds. The predicted isomer is essentially iso-energetic with the known Au25(SR)18- structure, but has a distinctly different optical spectrum. It has a significantly larger collision cross-section as compared to that of the known structure, which suggests it could be detectable in gas phase ion-mobility mass spectrometry.