A microscopic investigation of the rotational properties in ${}^{129}\mathrm{Cs}$ and ${}^{131}\mathrm{La}$ was carried out using the three-dimensional tilted axis cranking covariant density-functional theory (3DTAC-CDFT). The calculations reveal the coexistence of magnetic and chiral rotation built on the same quasiparticle configurations $\pi h_{11/2}^1\otimes \nu h_{11/2}^{-2}$ in ${}^{129}\mathrm{Cs}$ and ${}^{131}\mathrm{La}$, establishing a new type of shape coexistence. The calculations predict the deformation parameters for magnetic rotation in ${}^{129}\mathrm{Cs}$ ($\beta \approx 0.23, \gamma \approx {41}^{\circ }$) and ${}^{131}\mathrm{La}$ ($\beta \approx 0.25, \gamma \approx {42}^{\circ }$), along with those for possible chiral rotation in ${}^{129}\mathrm{Cs}$ ($\beta \approx 0.20, \gamma \approx {29}^{\circ }$) and in ${}^{131}\mathrm{La}$ ($\beta \approx 0.20, \gamma \approx {27}^{\circ }$). The energy spectra, the relationship between the spin and the rotational frequency, and the reduced $M1$ and $E2$ transition probabilities are obtained for various configurations. The experimental characteristics of band B8 in ${}^{129}\mathrm{Cs}$ and band 13 in ${}^{131}\mathrm{La}$ are reasonably reproduced. Moreover, a distinctive rotational-mode transition is uncovered in this work, progressing from principal-axis rotation to planar rotation and finally to chiral rotation as the rotational frequency evolves through different regimes.

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• Collective levels
• Electromagnetic transitions
• Nuclear many-body theory
• Nuclear structure & decays
• 90 ≤ A ≤ 149
• Nuclear density functional theory