ABSTRACT Constraining the escape fraction of Lyman Continuum (LyC) photons from high-redshift galaxies is crucial to understanding reionization. Recent observations have demonstrated that various characteristics of the Ly$\, \alpha$ emission line correlate with the inferred LyC escape fraction ($f_{\rm esc}^{\rm LyC}$) of low-redshift galaxies. Using a data set of 9600 mock Ly$\, \alpha$ spectra of star-forming galaxies at $4.64 \le z \le 6$ from the SPHINX$^{20}$ cosmological radiation hydrodynamical simulation, we study the physics controlling the escape of Ly$\, \alpha$ and LyC photons. We find that our mock Ly$\, \alpha$ observations are representative of high-redshift observations and that typical observational methods tend to overpredict the Ly$\, \alpha$ escape fraction ($f_{\rm esc}^{\rm Ly\, \alpha }$) by as much as 2 dex. We investigate the correlations between $f_{\rm esc}^{\rm LyC}$ and $f_{\rm esc}^{\rm Ly\, \alpha }$, Ly$\, \alpha$ equivalent width ($W_{\lambda }({\rm Ly\, \alpha })$), peak separation ($v_{\rm sep}$), central escape fraction ($f_{\rm cen}$), and red peak asymmetry ($A_f^{\rm red}$). We find that $f_{\rm esc}^{\rm Ly\, \alpha }$ and $f_{\rm cen}$ are good diagnostics for LyC leakage, selecting for galaxies with lower neutral gas densities and less UV attenuation that have recently experienced supernova feedback. In contrast, $W_{\lambda }({\rm Ly\, \alpha })$ and $v_{\rm sep}$ are found to be necessary but insufficient diagnostics, while $A_f^{\rm red}$ carries little information. Finally, we use stacks of Ly$\, \alpha$, H$\, \alpha$, and F150W mock surface brightness profiles to find that galaxies with high $f_{\rm esc}^{\rm LyC}$ tend to have less extended Ly$\, \alpha$ and F150W haloes but larger H$\, \alpha$ haloes than their non-leaking counterparts. This confirms that Ly$\, \alpha$ spectral profiles and surface brightness morphology can be used to better understand the escape of LyC photons from galaxies during the epoch of reionization.