The variety of shapes and structures, observed in light rare earth $A \sim 130$ nuclei, have been discussed in view of different angular momentum coupling schemes and their interplay that comes into effect at high spin. The $N = 79$ and 80 isotopes in La–Ce region, produced via fusion evaporation reaction, have been studied using the Indian National Gamma Array (INGA) consisting of 18 clover HPGe detectors. Two nearly degenerate $\Delta I = 1$ bands have been observed at high spin of ¹³⁷Ce and a triaxial deformation of $\gamma = \pm 30^{\circ}$ has been assigned to the bands, from the total Routhian surface (TRS) calculations. The high-spin candidates of the yrast band of ¹³⁸Ce show signature splitting both in energy and $B(M1)/B(E2)$ values. The bandcrossing due to the alignment of a pair of $h_{11}/2$ proton particles has been conjectured at $\hbar ω \sim 0.3$ MeV, from the single-particle Routhians obtained from TRS calculations. Lifetime measurements by Doppler shift attenuation method (DSAM) has been carried out and from the estimated reduced transition probability $B(M 1)$, the $\Delta I = 1$ band in ¹³⁸Ce has been characterized as a magnetic rotation (MR) band. The rise in the values of $B(M1)$, for the higher spin candidates of the band, has been conjectured as the reopening of a different shear at the top of the Band B1. The characteristic of the MR bands in $A \sim 130$ region has been discussed in the light of a phenomenological calculation and compared to the MR bands in other mass regions.