In an earlier paper, we presented the first evidence for a bow-shock nebula surrounding the X-ray binary LMC X-1 on a scale of ~15 pc, which we argued was powered by a jet associated with an accretion disk. We now present the first evidence for an ionization cone extending from an X-ray binary, a phenomenon only seen to date in active galactic nuclei (AGN). The ionization cone, detected in the HeII4686/Hbeta and [OIII]5007/Hbeta line ratio maps, aligns with the direction of the jet inferred from the bow-shock nebula. The cone has an opening angle ~45 deg and radial extent ~3.8 pc. Since the HeII emission cannot be explained by the companion O star, the gas in the ionization cone must be exposed to the `naked accretion disk, thereby allowing us to place constraints on the unobservable ionizing spectrum. The energetics of the ionization cone give unambiguous evidence for an ultraviolet - soft X-ray (XUV) excess in LMC X-1. Any attempt to match the hard X-ray spectrum (>1keV) with a conventional model of the accretion disk fails to account for this XUV component. We propose two likely sources for the observed anisotropy: (1) obscuration by a dusty torus, or (2) a jet-blown hole in a surrounding envelope of circumstellar absorbing material. We discuss the implications of our discovery in the context of the mass-scaling hypothesis for accretion onto black holes and suggest avenues for future research.