The Gould Belt Legacy Survey will map star-forming regions within 500 pc, using Heterodyne Array Receiver Programme (HARP), Submillimetre Common-User Bolometer Array 2 (SCUBA-2) and Polarimeter 2 (POL-2) on the James Clerk Maxwell Telescope (JCMT). This paper describes HARP observations of the J = 3 -> 2 transitions of 13CO and C18O towards Orion A. The 15 arcsec resolution observations cover 5 pc of the Orion filament, including OMC 1 (including BN-KL and Orion bar), OMC 2/3 and OMC 4, and allow a comparative study of the molecular gas properties throughout the star-forming cloud. The filament shows a velocity gradient of similar to 1 km s-1 pc-1 between OMC 1, 2 and 3, and high-velocity emission is detected in both isotopologues. The Orion Nebula and Bar have the largest masses and linewidths, and dominate the mass and energetics of the high-velocity material. Compact, spatially resolved emission from CH3CN, (CH3OH)-C-13, SO, HCOOCH3, CH3CHO and CH3OCHO is detected towards the Orion Hot Core. The cloud is warm, with a median excitation temperature of similar to 24 K; the Orion Bar has the highest excitation temperature gas, at >80 K. The (CO)-O-18 excitation temperature correlates well with the dust temperature (to within 40 per cent). The (CO)-O-18 emission is optically thin, and the (CO)-C-13 emission is marginally optically thick; despite its high mass, OMC 1 shows the lowest opacities. A virial analysis indicates that Orion A is too massive for thermal or turbulent support, but is consistent with a model of a filamentary cloud that is threaded by helical magnetic fields. The variation of physical conditions across the cloud is reflected in the physical characteristics of the dust cores. We find similar core properties between starless and protostellar cores, but variations in core properties with position in the filament. The OMC 1 cores have the highest velocity dispersions and masses, followed by OMC 2/3 and OMC 4. The differing fragmentation of these cores may explain why OMC 1 has formed clusters of high-mass stars, whereas OMC 4 produces fewer, predominantly low-mass stars.