We present an investigation of the interdependence of the optical-to-X-ray spectral slope ($alpha_{rm ox}$), the HeII equivalent-width (EW), and the monochromatic luminosity at 2500 Angstroms ($L_{2500}$). The values of $alpha_{rm ox}$ and HeII EW are indicators of the strength/shape of the quasar ionizing continuum, from the ultraviolet (UV; 1500--2500 Angstroms), through the extreme ultraviolet (EUV; 300--50 Angstroms), to the X-ray (2 keV) regime. For this investigation, we measure the HeII EW of 206 radio-quiet quasars devoid of broad absorption lines that have high-quality spectral observations of the UV and 2 keV X-rays. The sample spans wide redshift ($approx$ 0.13--3.5) and luminosity (log$(L_{2500}$) $approx$ 29.2--32.5 erg s$^{-1}$ Hz$^{-1}$) ranges. We recover the well-known $alpha_{rm ox}$--$L_{2500}$ and HeII EW--$L_{2500}$ anti-correlations, and we find a similarly strong correlation between $alpha_{rm ox}$ and HeII EW, and thus the overall spectral shape from the UV, through the EUV, to the X-ray regime is largely set by luminosity. A significant $alpha_{rm ox}$--HeII EW correlation remains after removing the contribution of $L_{2500}$ from each quantity, and thus the emission in the EUV and the X-rays are also directly tied. This set of relations is surprising, since the UV, EUV, and X-ray emission are expected to be formed in three physically distinct regions. Our results indicate the presence of a redshift-independent physical mechanism that couples the continuum emission from these three different regions, and thus controls the overall continuum shape from the UV to the X-ray regime.
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