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We present ALMA observations of the $^{12}$CO, $^{13}$CO, C$^{18}$O J=2-1 transitions and the 1.3\,mm continuum emission for the circumbinary disc around HD 142527, at an angular resolution of $\approx$\,0\farcs3. We observe multiple spiral structures in intensity, velocity and velocity dispersion for the $^{12}$CO and $^{13}$CO gas tracers. A newly detected $^{12}$CO spiral originates from the dust horseshoe, and is rotating at super-Keplerian velocity or vertically ascending, whilst the inter-spiral gas is rotating at sub-Keplerian velocities. This new spiral possibly connects to a previously identified spiral, thus spanning > 360$^\circ$. A spatial offset of ~30 au is observed between the $^{12}$CO and $^{13}$CO spirals, to which we hypothesize that the gas layers are propagating at different speeds (``surfing'') due to a non-zero vertical temperature gradient. Leveraging the varying optical depths between the CO isotopologues, we reconstruct temperature and column density maps of the outer disc. Gas surface density peaks at r\,$\approx$\,180\,au, coincident with the peak of continuum emission. Here the dust grains have a Stokes number of $\approx$\,1, confirming radial and azimuthal trapping in the horseshoe. We measure a cavity radius at half-maximum surface density of $\approx$\,100\,au, and a cavity eccentricity between 0.3 and 0.45.