Understanding how crystal symmetry governs phonon-phonon scattering is essential for predicting heat transport in two-dimensional semiconductors. While most studies focus on three-phonon (3ph) interactions, growing evidence shows that four-phonon (4ph) processes can be equally important for thermal transport, particularly when phonon scattering is constrained by crystal symmetry beyond energy and momentum conservation. In this work, we investigate the role of out-of-plane reflection symmetry by comparing two phases of monolayer ${\mathrm{HfH}}_2$, which exhibit nearly identical harmonic properties but distinct symmetry. Their large acoustic-optical band gap suppresses scattering channels in which two or three acoustic phonons combine into an optical phonon, thereby enabling a clean isolation of higher-order scattering governed by reflection symmetry. By combining machine-learning interatomic potentials with group theoretical selection-rule analysis, we show that the 1H phase supports symmetry-constrained even-flexural-acoustic (ZA) 4ph channels that greatly enhance higher-order anharmonicity. In contrast, the 1T phase, which lacks out-of-plane reflection symmetry, remains dominated by 3ph scattering with only minor 4ph contributions. As a result, at 300 K the activation of symmetry-constrained channels in the 1H phase reduces the lattice thermal conductivity by more than half and strongly suppresses the ZA-mode contribution. Our work establishes out-of-plane reflection symmetry as a key enabler of strong 4ph scattering and suggests that symmetry engineering may provide a viable route to controlling phonon thermal transport in other two-dimensional transition-metal semiconductors with large acoustic-optical band gaps.

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• Anharmonic lattice dynamics
• Lattice thermal conductivity
• Thermal transport
• 2-dimensional systems
• Layered semiconductors
• Ab initio molecular dynamics
• Boltzmann theory
• Density functional calculations