  The Ly$\alpha$ line is a powerful probe of distant galaxies, which contains
information about inflowing/outflowing gas through which Ly$\alpha$ photons
scatter. To develop our understanding of this probe, we post-process a zoom-in
radiation-hydrodynamics simulation of a low-mass ($M_* \sim 10^9 M_\odot$)
galaxy to construct 22500 mock spectra in 300 directions from $z = 3$ to 4.
Remarkably, we show that one galaxy can reproduce the variety of a large sample
of spectroscopically observed Ly$\alpha$ line profiles. While most mock spectra
exhibit double-peak profiles with a dominant red peak, their shapes cover a
large parameter space in terms of peak velocities, peak separation and flux
ratio. This diversity originates from radiative transfer effects at ISM and CGM
scales, and depends on galaxy inclination and evolutionary phase. Red-dominated
lines preferentially arise in face-on directions during post-starburst outflows
and are bright. Conversely, accretion phases usually yield symmetric double
peaks in the edge-on direction and are fainter. While resonant scattering
effects at $< 0.2\times R_{\rm vir}$ are responsible for the broadening and
velocity shift of the red peak, the extended CGM acts as a screen and impacts
the observed peak separation. The ability of simulations to reproduce observed
Ly$\alpha$ profiles and link their properties with galaxy physical parameters
offers new perspectives to use Ly$\alpha$ to constrain the mechanisms that
regulate galaxy formation and evolution. Notably, our study implies that deeper
Ly$\alpha$ surveys may unveil a new population of blue-dominated lines tracing
inflowing gas.
