We present spatial analysis of the deep (57 ks) ROSAT HRI X-ray image of the E4 galaxy NGC 720. The orientation of the HRI surface brightness is consistent with the optical position angle (P.A.) interior to semimajor axis a similar to 60 '' (optical R(e) similar to 50 ''). For larger a, the isophotes twist and eventually (a greater than or similar to 100 '') orient along a direction consistent with the P.A. measured with the PSPC data (Buote & Canizares)- the similar to 30 degrees twist is significant at an estimated 99% confidence level. We argue that this twist is not the result of projected foreground and background sources, ram pressure effects, or tidal distortions. If spheroidal symmetry and a nearly isothermal hot gas are assumed, then the azimuthally averaged radial profile displays features that, when combined with the observed P.A. twist, are inconsistent with the simple assumptions that the X-ray emission arises either entirely from hot gas or from the combined emission of hot gas and discrete sources. We discuss possible origins of the P.A. twist and radial profile features (e.g., triaxiality).