To map the celestial hemisphere, we decided to use stereographic projection, as we have found that most historical star charts use it or something very similar, nevertheless, we could have picked any of the dozens of other projection techniques. The point being, choosing a map projection and its parameters is mostly a question of "taste" as well as utilitarian considerations of the map creator. Examine this "unusual" projection of the Earth for example: McArthur's Earth
Here's a very crude illustration of typical stereographic celestial projections:
Based on early feedback on our project, we decided to switch to a different projection aspect, as such:
We elected to switch the aspect point to a zenith point above the chosen location because we have found that the mirrored position of cardinal directions is confusing for the eye that is used to North -> East -> South -> West (CW), especially considering how prominent these directions are in our design. With the new aspect, we could keep the "traditional" order that most people are used to on maps of the Earth while keeping the projection factual.
The projection we use is true, but the aspect is different. You can think of it as "the celestial hemisphere looks from above", or, sentimentally put, the view from Heaven. Indeed, not a typical choice of projection.
In our projection, we have chosen not to show the Moon or the Sun.
Fun fact (thanks to our customer Chris B): This view is similar to that of the atlas produced by Johannes Hevelius in 1690. All of the constellations are drawn from the perspective outside the celestial sphere. Hence, they appear reversed. See Uranographia by Johannes Hevelius.