We investigate 3D atmosphere dynamics for tidally locked terrestrial planets with an Earth-like atmosphere and irradiation for different rotation periods (P rot = 1–100 d) and planet sizes (R P = 1–2R Earth) with unprecedented fine detail. We could precisely identify three climate state transition regions that are associated with phase transitions in standing tropical and extratropical Rossby waves. We confirm that the climate on fast-rotating planets may assume multiple states (P rot ≤ 12 d for R P = 2R Earth). Our study is, however, the first to identify the type of planetary wave associated with different climate states: the first state is dominated by standing tropical Rossby waves with fast equatorial superrotation. The second state is dominated by standing extratropical Rossby waves with high-latitude westerly jets with slower wind speeds. For very fast rotations (P rot ≤ 5 d for R P = 2R Earth), we find another climate state transition, where the standing tropical and extratropical Rossby wave can both fit on the planet. Thus, a third state with a mixture of the two planetary waves becomes possible that exhibits three jets. Different climate states may be observable, because the upper atmosphere's hotspot is eastward shifted with respect to the substellar point in the first state, westward shifted in the second state and the third state shows a longitudinal ‘smearing’ of the spot across the substellar point. We show, furthermore, that the largest fast-rotating planet in our study exhibits atmosphere features known from hot Jupiters like fast equatorial superrotation and a temperature chevron in the upper atmosphere.