There is an intrinsic value in higher integration of multi-carrier energy systems (especially gas and electricity), to increase operational flexibility in the electricity system and to improve allocation of resources in gas and electricity networks. The integration of different energy carrier markets is challenging due to the existence of physical and economic dependencies between the different energy carriers. We propose in this paper an integrated day-ahead multi-carrier gas, electricity and heat market clearing which includes new types of orders and constraints on these orders to represent techno-economic constraints of con-version and storage technologies. We prove that the proposed market clearing gives rise to competitive equilibria. In addition, we propose two decentralised clearing algorithms which differ in how the decomposition of the underlying centralised clearing optimisation problem is performed. This has implications in terms of the involved agents and their mutual information exchange. It is proven that they yield solutions equivalent to the centralised market clearing under a mild assumption of sufficient number of iterations. We argue that such an integrated multi-carrier energy market mitigates (spot) market risks faced by market participants and enables better spot pricing of the different energy carriers. The results show that conversion/storage technology owners would suffer from losses and/or opportunity costs, if they were obliged to only use elementary orders. For the test cases considered in this article, sum of losses and opportunity costs could reach up to 13,000 €/day and 9,000 €/day respectively, compared with the case where conversion and storage orders are used.