  We start by constructing a conformally covariant improvement of the celestial
light transform which keeps track of the mixing between incoming and outgoing
states under finite Lorentz transformations in $\mathbb{R}^{2,2}$. We then
compute generic 2, 3 and 4-point celestial amplitudes for massless external
states in the ambidextrous basis prepared by composing this
$\mathrm{SL}(2,\mathbb{R})$ intertwiner with the usual celestial map between
momentum and boost eigenstates. The results are non-distributional in the
celestial coordinates $(z,\bar{z})$ and conformally covariant in all scattering
channels. Finally, we focus on the tree level 4-gluon amplitude where we
present a streamlined route to the ambidextrous correlator based on
Grassmannian formulae and examine its alpha space representation. In the
process, we gain insights into the operator dictionary and CFT data of the
holographic dual.
