The extremely narrow natural linewidths of rovibrational energy levels in the molecular hydrogen ion ${\mathrm{H}}_2^{+}$, and the prospect of synthesising its antimatter counterpart ${\overline{\mathrm{H}}}_2^{-}$, make it a promising candidate for high-precision tests of fundamental symmetries such as Lorentz and $CPT$ invariance. In this paper, we present a detailed analysis of the rovibrational spectrum of the (anti)hydrogen molecular ion in a low-energy effective theory incorporating Lorentz and $CPT$ violation. The focus is on the spin-independent couplings in this theory, and especially the $CPT$ odd couplings for which the best current bounds come from measurements of the $1S\text{−}2S$ transition in atomic hydrogen and antihydrogen. We show that in addition to the improvement in these bounds from the increased precision of the transition frequencies, potentially reaching 1 part in ${10}^{17}$, rovibrational transitions in the ${\mathrm{H}}_2^{+}$ and ${\overline{\mathrm{H}}}_2^{-}$ molecular ions have an enhanced sensitivity to Lorentz and $CPT$ violation of $O(m_p/m_e)$ in the proton (hadron) sector compared to $\mathrm{H}$ and $\overline{\mathrm{H}}$ atomic transitions.

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• Field & string theory models & techniques
• Symmetries