We present an investigation of the interdependence of the optical-to-X-ray spectral slope (aox), the HeII equivalent-width (EW), and the monochromatic luminosity at 2500{AA} (L2500). The values of {alpha}_ox_ and HeII EW are indicators of the strength/shape of the quasar ionizing continuum, from the ultraviolet (UV; 1500-2500{AA}), through the extreme ultraviolet (EUV;300-50{AA}), to the X-ray (2keV) 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 2keV X-rays. The sample spans wide redshift (z~0.13-3.5) and luminosity (logL_2500_~29.2-32.5erg/s/Hz) ranges. We recover the well-known {alpha}_ox_-L_2500_ and HeII EW-L_2500_ anti-correlations, and we find a similarly strong correlation between {alpha}_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}_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.