We present results from a deep Suzaku observation of the black hole in LMC X-1, supplemented by coincident monitoring with the Rossi X-ray Timing Explorer (RXTE). We identify broad relativistic reflection features in a soft disc-dominated spectrum. A strong and variable power-law component of emission is present which we use to demonstrate that enhanced Comptonization strengthens disc reflection. We constrain the spin parameter of the black hole by modelling LMC X-1's broad reflection features. For our primary and most comprehensive spectral model, we obtain a high value for the spin: (68 per cent confidence). However, by additionally considering two alternate models as a measure of our systematic uncertainty, we obtain a broader constraint: . Both of these spin values are entirely consistent with a previous estimate of spin obtained using the continuum-fitting method. At 99 per cent confidence, the reflection features require a* > 0.2. In addition to modelling the relativistically broadened reflection, we also model a sharp and prominent reflection component that provides strong evidence for substantial reprocessing in the wind of the massive companion. We infer that this wind sustains the ionization cone surrounding the binary system; this hypothesis naturally produces appropriate and consistent mass, time and length scales for the cone structure.