  We present the first statistical analysis of kinematically-resolved,
spatially-extended Ly$\alpha$ emission around $z = 2-3$ galaxies in the Keck
Baryonic Structure Survey (KBSS) using the Keck Cosmic Web Imager (KCWI). Our
sample of 59 star-forming galaxies ($z_\mathrm{med} = 2.29$) comprises the
subset with typical KCWI integration times of ~5 hours and with existing
imaging data from the Hubble Space Telescope and/or adaptive optics-assisted
integral field spectroscopy. The high resolution images were used to evaluate
the azimuthal dependence of the diffuse Ly$\alpha$ emission with respect to the
stellar continuum within projected galactocentric distances of $\lesssim 30$
proper kpc. We introduce cylindrically-projected 2D spectra (CP2D) that map the
averaged Ly$\alpha$ spectral profile over a specified range of azimuthal angle,
as a function of impact parameter around galaxies. The averaged CP2D spectrum
of all galaxies shows clear signatures of Ly$\alpha$ resonant scattering by
outflowing gas. We stacked the CP2D spectra of individual galaxies over ranges
of azimuthal angle with respect to their major axes. The extended Ly$\alpha$
emission along the galaxy principal axes are statistically indistinguishable,
with residual asymmetry of $\le$ 2% ($\sim 2 \sigma$) of the integrated
Ly$\alpha$ emission. The symmetry implies that the Ly$\alpha$ scattering medium
is dominated by outflows in all directions within 30 kpc. Meanwhile, we find
that the blueshifted component of Ly$\alpha$ emission is marginally stronger
along galaxy minor axes for galaxies with relatively weak Ly$\alpha$ emission.
We speculate that this weak directional dependence of Ly$\alpha$ emission
becomes discernible only when the Ly$\alpha$ escape fraction is low. These
discoveries highlight the need for similar analyses in simulations with
Ly$\alpha$ radiative transfer modeling.
