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Multienergy system (MES) operation has proven beneficial to the flexibility and efficiency of energy usage. Although efforts have been made toward the simulation and optimization of MESs, successful MES deployment is a grand challenge for market operations. The previous literature focuses on the modeling of the MES and the formulation of corresponding locational marginal prices (LMPs). A detailed analysis of LMP properties in an MES market-clearing model has not been fully developed. Although coupling analysis between different energy systems has been conducted from a system and power flow perspective, a coupling analysis with a specific indicator from a LMP perspective could provide operators with new insight into MES integration. Therefore, this paper first identifies six properties of the LMP in a unified MES market-clearing model. Then, based on the identified properties, two new types of critical load levels (CLL), Self-CLL and Cross-CLL, are defined and discussed to investigate the MES price pattern. Next, the Self-CLLs and Cross-CLLs for an MES are quantified by grey relation analysis (GRA) to deliver an MES coupling analysis from the perspective of LMP patterns. The identified properties and proposed analysis are demonstrated with a small MES case and a large one.