We point out that the rotation angle $\beta$ of cosmic birefringence, which is a recently reported parity-violating signal in the cosmic microwave background (CMB), has a phase ambiguity of $n\pi (n\in \mathbb{Z})$. This ambiguity has a significant impact on the interpretation of the origin of cosmic birefringence. Assuming an axionlike particle (ALP) as the origin of cosmic birefringence, this ambiguity can be partly broken by the anisotropic cosmic birefringence and the shape of the CMB angular power spectra. We forecast constraints on $n$ from ongoing and upcoming experiments and find that most nonzero values of $n$ can be excluded using the cross-power spectrum between $E$ and $B$ modes. We also find that measuring the reionization bump in the $E$-mode auto-power spectrum can give further constraints, provided that the optical depth $\tau$ is determined independently of the CMB. The change in the low-$l$ $E$-mode power spectrum may also provide a potential solution to the tension in $\tau$ suggested by the latest CMB and baryon acoustic oscillation measurements.

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• Cosmic microwave background
• Cosmology
• Dark matter