CeAlGe, a proposed type-II Weyl semimetal, orders antiferromagnetically below 5 K. Both a spin-flop and a spin-flip transitions to less than 1 $mu_B$/Ce are observed at 2 K below 30 kOe in the $M(H)$ ($bf{H}|bf{a}$ and $bf{b}$) and 4.3 kOe ($bf{H}|langle110rangle$) data, respectively, indicating a four-fold symmetry of the $M(H)$ along the principal directions in the tetragonal $it{ab}$-plane with $langle110rangle$ set of easy directions. However, anomalously robust and complex two-fold symmetry is observed in the angular dependence of resistivity and magnetic torque data in the magnetically ordered state once the field is swept in the $it{ab}$-plane. This two-fold symmetry is independent of temperature- and field-hystereses and suggests a magnetic phase transition that separates two different magnetic structures in the $it{ab}$-plane. The boundary of this magnetic phase transition can be tuned by different growth conditions.
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