The inner shape can't be symmetrical if you want an asymmetric pattern but if you want then to both have the same pattern control frequency the final size must be the same or very similar.
I'm not aware of anyone claiming to do it for that reason, the floor is usually considered to be less detrimental to timbre modification than the ceiling so I'm not sure it would work out well for that reason alone.
JinMTVT, I recently started a thread about tilting back speakers, wrote some things that happen and probably there is more to it given what kind of speaker you have. To give you starter what happens with tilt. https://www.diyaudio.com/community/...ymmetric-crossover-and-tilt-experiment.388389
I haven't made any comparative tests of different speaker directivities. I have run a lot of simulations that show what sort of directivity will result from different concepts. I can see the benefit of controlling the rear radiation particularly in the 200 to 500Hz range where there can be a lot of negative SBIR interaction in typical placements. Below that getting higher directivity becomes increasingly difficult and it is then becomes situation dependent on what is the best practical approach.