  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.
